- 7.1 Solving Trigonometric Equations with Identities
- Introduction to Functions
- 1.1 Functions and Function Notation
- 1.2 Domain and Range
- 1.3 Rates of Change and Behavior of Graphs
- 1.4 Composition of Functions
- 1.5 Transformation of Functions
- 1.6 Absolute Value Functions
- 1.7 Inverse Functions
- Key Equations
- Key Concepts
- Review Exercises
- Practice Test
- Introduction to Linear Functions
- 2.1 Linear Functions
- 2.2 Graphs of Linear Functions
- 2.3 Modeling with Linear Functions
- 2.4 Fitting Linear Models to Data
- Introduction to Polynomial and Rational Functions
- 3.1 Complex Numbers
- 3.2 Quadratic Functions
- 3.3 Power Functions and Polynomial Functions
- 3.4 Graphs of Polynomial Functions
- 3.5 Dividing Polynomials
- 3.6 Zeros of Polynomial Functions
- 3.7 Rational Functions
- 3.8 Inverses and Radical Functions
- 3.9 Modeling Using Variation
- Introduction to Exponential and Logarithmic Functions
- 4.1 Exponential Functions
- 4.2 Graphs of Exponential Functions
- 4.3 Logarithmic Functions
- 4.4 Graphs of Logarithmic Functions
- 4.5 Logarithmic Properties
- 4.6 Exponential and Logarithmic Equations
- 4.7 Exponential and Logarithmic Models
- 4.8 Fitting Exponential Models to Data
- Introduction to Trigonometric Functions
- 5.2 Unit Circle: Sine and Cosine Functions
- 5.3 The Other Trigonometric Functions
- 5.4 Right Triangle Trigonometry
- Introduction to Periodic Functions
- 6.1 Graphs of the Sine and Cosine Functions
- 6.2 Graphs of the Other Trigonometric Functions
- 6.3 Inverse Trigonometric Functions
- Introduction to Trigonometric Identities and Equations
- 7.2 Sum and Difference Identities
- 7.3 Double-Angle, Half-Angle, and Reduction Formulas
- 7.4 Sum-to-Product and Product-to-Sum Formulas
- 7.5 Solving Trigonometric Equations
- 7.6 Modeling with Trigonometric Functions
- Introduction to Further Applications of Trigonometry
- 8.1 Non-right Triangles: Law of Sines
- 8.2 Non-right Triangles: Law of Cosines
- 8.3 Polar Coordinates
- 8.4 Polar Coordinates: Graphs
- 8.5 Polar Form of Complex Numbers
- 8.6 Parametric Equations
- 8.7 Parametric Equations: Graphs
- 8.8 Vectors
- Introduction to Systems of Equations and Inequalities
- 9.1 Systems of Linear Equations: Two Variables
- 9.2 Systems of Linear Equations: Three Variables
- 9.3 Systems of Nonlinear Equations and Inequalities: Two Variables
- 9.4 Partial Fractions
- 9.5 Matrices and Matrix Operations
- 9.6 Solving Systems with Gaussian Elimination
- 9.7 Solving Systems with Inverses
- 9.8 Solving Systems with Cramer's Rule
- Introduction to Analytic Geometry
- 10.1 The Ellipse
- 10.2 The Hyperbola
- 10.3 The Parabola
- 10.4 Rotation of Axes
- 10.5 Conic Sections in Polar Coordinates
- Introduction to Sequences, Probability and Counting Theory
- 11.1 Sequences and Their Notations
- 11.2 Arithmetic Sequences
- 11.3 Geometric Sequences
- 11.4 Series and Their Notations
- 11.5 Counting Principles
- 11.6 Binomial Theorem
- 11.7 Probability
- Introduction to Calculus
- 12.1 Finding Limits: Numerical and Graphical Approaches
- 12.2 Finding Limits: Properties of Limits
- 12.3 Continuity
- 12.4 Derivatives
- A | Basic Functions and Identities

## Learning Objectives

- Verify the fundamental trigonometric identities.
- Simplify trigonometric expressions using algebra and the identities.

## Verifying the Fundamental Trigonometric Identities

Prove: 1 + cot 2 θ = csc 2 θ 1 + cot 2 θ = csc 2 θ

Recall that an even function is one in which

- Since sin (− θ ) = − sin θ , sin (− θ ) = − sin θ , sine is an odd function.
- Since, cos (− θ ) = cos θ , cos (− θ ) = cos θ , cosine is an even function.

## Summarizing Trigonometric Identities

The Pythagorean Identities are based on the properties of a right triangle.

The reciprocal identities define reciprocals of the trigonometric functions.

The quotient identities define the relationship among the trigonometric functions.

## Graphing the Equations of an Identity

Given a trigonometric identity, verify that it is true.

- Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to simplify than to build.
- Look for opportunities to factor expressions, square a binomial, or add fractions.
- Noting which functions are in the final expression, look for opportunities to use the identities and make the proper substitutions.
- If these steps do not yield the desired result, try converting all terms to sines and cosines.

## Verifying a Trigonometric Identity

Verify tan θ cos θ = sin θ . tan θ cos θ = sin θ .

We will start on the left side, as it is the more complicated side:

Verify the identity csc θ cos θ tan θ = 1. csc θ cos θ tan θ = 1.

## Verifying the Equivalency Using the Even-Odd Identities

Verify the following equivalency using the even-odd identities:

Working on the left side of the equation, we have

## Verifying a Trigonometric Identity Involving sec 2 θ

Verify the identity sec 2 θ − 1 sec 2 θ = sin 2 θ sec 2 θ − 1 sec 2 θ = sin 2 θ

As the left side is more complicated, let’s begin there.

Show that cot θ csc θ = cos θ . cot θ csc θ = cos θ .

## Creating and Verifying an Identity

## Verifying an Identity Using Algebra and Even/Odd Identities

Let’s start with the left side and simplify:

## Verifying an Identity Involving Cosines and Cotangents

Verify the identity: ( 1 − cos 2 x ) ( 1 + cot 2 x ) = 1. ( 1 − cos 2 x ) ( 1 + cot 2 x ) = 1.

We will work on the left side of the equation.

## Using Algebra to Simplify Trigonometric Expressions

## Writing the Trigonometric Expression as an Algebraic Expression

## Rewriting a Trigonometric Expression Using the Difference of Squares

Rewrite the trigonometric expression: 4 cos 2 θ − 1. 4 cos 2 θ − 1.

Rewrite the trigonometric expression: 25 − 9 sin 2 θ . 25 − 9 sin 2 θ .

## Simplify by Rewriting and Using Substitution

Simplify the expression by rewriting and using identities:

We can start with the Pythagorean identity.

Now we can simplify by substituting 1 + cot 2 θ 1 + cot 2 θ for csc 2 θ . csc 2 θ . We have

(Hint: Multiply the numerator and denominator on the left side by 1 − sin θ . ) 1 − sin θ . )

## 7.1 Section Exercises

For the following exercises, use the fundamental identities to fully simplify the expression.

sin x cos x sec x sin x cos x sec x

sin ( − x ) cos ( − x ) csc ( − x ) sin ( − x ) cos ( − x ) csc ( − x )

tan x sin x + sec x cos 2 x tan x sin x + sec x cos 2 x

csc x + cos x cot ( − x ) csc x + cos x cot ( − x )

cot t + tan t sec ( − t ) cot t + tan t sec ( − t )

3 sin 3 t csc t + cos 2 t + 2 cos ( − t ) cos t 3 sin 3 t csc t + cos 2 t + 2 cos ( − t ) cos t

− tan ( − x ) cot ( − x ) − tan ( − x ) cot ( − x )

− sin ( − x ) cos x sec x csc x tan x cot x − sin ( − x ) cos x sec x csc x tan x cot x

1 + tan 2 θ csc 2 θ + sin 2 θ + 1 sec 2 θ 1 + tan 2 θ csc 2 θ + sin 2 θ + 1 sec 2 θ

1 − cos 2 x tan 2 x + 2 sin 2 x 1 − cos 2 x tan 2 x + 2 sin 2 x

tan x + cot x csc x ; cos x tan x + cot x csc x ; cos x

sec x + csc x 1 + tan x ; sin x sec x + csc x 1 + tan x ; sin x

cos x 1 + sin x + tan x ; cos x cos x 1 + sin x + tan x ; cos x

1 sin x cos x − cot x ; cot x 1 sin x cos x − cot x ; cot x

1 1 − cos x − cos x 1 + cos x ; csc x 1 1 − cos x − cos x 1 + cos x ; csc x

1 csc x − sin x ; sec x and tan x 1 csc x − sin x ; sec x and tan x

For the following exercises, verify the identity.

cos x − cos 3 x = cos x sin 2 x cos x − cos 3 x = cos x sin 2 x

cos x ( tan x − sec ( − x ) ) = sin x − 1 cos x ( tan x − sec ( − x ) ) = sin x − 1

( sin x + cos x ) 2 = 1 + 2 sin x cos x ( sin x + cos x ) 2 = 1 + 2 sin x cos x

cos 2 x − tan 2 x = 2 − sin 2 x − sec 2 x cos 2 x − tan 2 x = 2 − sin 2 x − sec 2 x

For the following exercises, prove or disprove the identity.

1 1 + cos x − 1 1 − cos ( − x ) = − 2 cot x csc x 1 1 + cos x − 1 1 − cos ( − x ) = − 2 cot x csc x

csc 2 x ( 1 + sin 2 x ) = cot 2 x csc 2 x ( 1 + sin 2 x ) = cot 2 x

tan x sec x sin ( − x ) = cos 2 x tan x sec x sin ( − x ) = cos 2 x

sec ( − x ) tan x + cot x = − sin ( − x ) sec ( − x ) tan x + cot x = − sin ( − x )

1 + sin x cos x = cos x 1 + sin ( − x ) 1 + sin x cos x = cos x 1 + sin ( − x )

cos 2 θ − sin 2 θ 1 − tan 2 θ = sin 2 θ cos 2 θ − sin 2 θ 1 − tan 2 θ = sin 2 θ

3 sin 2 θ + 4 cos 2 θ = 3 + cos 2 θ 3 sin 2 θ + 4 cos 2 θ = 3 + cos 2 θ

sec θ + tan θ cot θ + cos θ = sec 2 θ sec θ + tan θ cot θ + cos θ = sec 2 θ

As an Amazon Associate we earn from qualifying purchases.

Access for free at https://openstax.org/books/precalculus-2e/pages/1-introduction-to-functions

- Authors: Jay Abramson
- Publisher/website: OpenStax
- Book title: Precalculus 2e
- Publication date: Dec 21, 2021
- Location: Houston, Texas
- Book URL: https://openstax.org/books/precalculus-2e/pages/1-introduction-to-functions
- Section URL: https://openstax.org/books/precalculus-2e/pages/7-1-solving-trigonometric-equations-with-identities

## PROBLEMS ON TRIGONOMETRIC IDENTITIES WITH SOLUTIONS

Let A = (1 - cos 2 θ) csc 2 θ and B = 1.

Because sin 2 θ + cos 2 θ = 1, we have

Let A = sec θ √(1 - sin 2 θ) and B = 1.

Let A = tan θ sin θ + cos θ and B = sec θ.

A = (sin θ/cos θ) ⋅ sin θ + cos θ

A = (sin 2 θ /cos θ) + (cos 2 θ/cosθ)

A = (sin 2 θ + cos 2 θ ) / cos θ

(1 - cos θ)(1 + cos θ)(1 + cot 2 θ) = 1

Let A = (1 - cos θ)(1 + cos θ)(1 + cot 2 θ) = 1 and B = 1.

A = (1 - cos θ)(1 + cos θ)(1 + cot 2 θ)

A = (1 - cos 2 θ)(1 + cot 2 θ)

A = sin 2 θ + sin 2 θ ⋅ cot 2 θ

A = sin 2 θ + sin 2 θ ⋅ (cos 2 θ/sin 2 θ)

Let A = cot θ + tan θ and B = sec θ csc θ.

A = (cos θ/sin θ) + (sin θ/cos θ)

A = (cos 2 θ/sin θ cos θ) + (sin 2 θ/sin θ cos θ)

A = (cos 2 θ + sin 2 θ) / sin θ cos θ

cos θ/(1 - tan θ) + sin θ/(1 - cot θ) = sin θ + cos θ

Let A = cos θ/(1 - tan θ) + sin θ/(1 - cot θ) and

A = cos θ/{1 - (sin θ/cos θ)} + sin θ/{1 - (cos θ/sin θ)}

A = cos 2 θ /(cos θ - sin θ) + sin 2 θ /(sin θ - cos θ)

A = cos 2 θ /(cos θ - sin θ) - sin 2 θ /(cos θ - sin θ)

A = (cos 2 θ - sin 2 θ ) / (cos θ - sin θ)

A = [(cos θ + sin θ)(cos θ - sin θ)] / (cos θ - sin θ)

tan 4 θ + tan 2 θ = sec 4 θ - sec 2 θ

Let A = tan 4 θ + tan 2 θ and B = sec 4 θ + sec 2 θ.

√{(sec θ – 1)/(sec θ + 1)} = cosec θ - cot θ

Let A = √{(sec θ – 1)/(sec θ + 1)} and B = cosec θ - cot θ.

A = √{(sec θ – 1)/(sec θ + 1)}

A = √[{(sec θ - 1) (sec θ - 1)}/{(sec θ + 1) (sec θ - 1)}]

A = √{(sec θ - 1) 2 / (sec 2 θ - 1)}

A = √{(sec θ - 1) 2 / tan 2 θ }

A = {(1/cos θ)/(sin θ/cos θ)} - cot θ

A = {(1/cos θ) ⋅ (cos θ/sin θ)} - cot θ

(1 - sin A)/(1 + sin A) = (sec A - tan A) 2

Let A = (1 - sin A)/(1 + sin A) and B = (sec A - tan A) 2 .

A = (1 - sin A) 2 / (1 - sin A) (1 + sin A)

A = (1 - sin A) 2 / (1 - sin 2 A)

A = {(1/cos A) - (sin A/cos A)} 2

(tan θ + sec θ - 1)/(tan θ - sec θ + 1) = (1 + sin θ)/cos θ

Let A = (tan θ + sec θ - 1)/(tan θ - sec θ + 1) and

A = (tan θ + sec θ - 1)/(tan θ - sec θ + 1)

A = [(tan θ + sec θ) - (sec 2 θ - tan 2 θ )]/(tan θ - sec θ + 1)

A = {(tan θ + sec θ) (1 - sec θ + tan θ)}/(tan θ - sec θ + 1)

A = {(tan θ + sec θ) (tan θ - sec θ + 1)}/(tan θ - sec θ + 1)

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## Prime and Composite Numbers

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Proving the problems on trigonometric identities:

- Basic Trigonometric Ratios and Their Names
- Restrictions of Trigonometrical Ratios
- Reciprocal Relations of Trigonometric Ratios
- Quotient Relations of Trigonometric Ratios
- Limit of Trigonometric Ratios
- Trigonometrical Identity
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- Eliminate Theta between the equations
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- Trig Ratio Problems
- Proving Trigonometric Ratios
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- Problems on Trigonometric Ratio of Standard Angle
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- T rigonometrical Ratios of (270° - θ)
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- Trigonometrical Ratios of some Particular Angles
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## 9.1: Solving Trigonometric Equations with Identities

## Learning Objectives

- Verify the fundamental trigonometric identities.
- Simplify trigonometric expressions using algebra and the identities.

## Verifying the Fundamental Trigonometric Identities

Prove: \(1+{\cot}^2 \theta={\csc}^2 \theta\)

This is shown in Figure \(\PageIndex{2}\).

Recall that an even function is one in which

\(f(−x)=f(x)\) for all \(x\) in the domain of \(f\)

- Since \(\sin(−\theta)=−\sin \theta\),sine is an odd function.
- Since \(\cos(−\theta)=\cos \theta\),cosine is an even function.

\[\csc(−\theta)=\dfrac{1}{\sin(−\theta)}=\dfrac{1}{−\sin \theta}=−\csc \theta. \nonumber\]

The cosecant function is therefore odd.

\[\sec(−\theta)=\dfrac{1}{\cos(−\theta)}=\dfrac{1}{\cos \theta}=\sec \theta. \nonumber\]

The secant function is therefore even.

## SUMMARIZING TRIGONOMETRIC IDENTITIES

The Pythagorean identities are based on the properties of a right triangle.

\[{\cos}^2 \theta+{\sin}^2 \theta=1\]

\[1+{\cot}^2 \theta={\csc}^2 \theta\]

\[1+{\tan}^2 \theta={\sec}^2 \theta\]

\[\tan(−\theta)=−\tan \theta\]

\[\cot(−\theta)=−\cot \theta\]

\[\sin(−\theta)=−\sin \theta\]

\[\csc(−\theta)=−\csc \theta\]

The reciprocal identities define reciprocals of the trigonometric functions.

\[\sin \theta=\dfrac{1}{\csc \theta}\]

\[\cos \theta=\dfrac{1}{\sec \theta}\]

\[\tan \theta=\dfrac{1}{\cot \theta}\]

\[\csc \theta=\dfrac{1}{\sin \theta}\]

\[\sec \theta=\dfrac{1}{\cos \theta}\]

\[\cot \theta=\dfrac{1}{\tan \theta}\]

The quotient identities define the relationship among the trigonometric functions.

\[\tan \theta=\dfrac{\sin \theta}{\cos \theta}\]

\[\cot \theta=\dfrac{\cos \theta}{\sin \theta}\]

## Example \(\PageIndex{1}\): Graphing the Equations of an Identity

## How to: Given a trigonometric identity, verify that it is true.

- Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to simplify than to build.
- Look for opportunities to factor expressions, square a binomial, or add fractions.
- Noting which functions are in the final expression, look for opportunities to use the identities and make the proper substitutions.
- If these steps do not yield the desired result, try converting all terms to sines and cosines.

## Example \(\PageIndex{2}\): Verifying a Trigonometric Identity

Verify \(\tan \theta \cos \theta=\sin \theta\).

We will start on the left side, as it is the more complicated side:

## Exercise \(\PageIndex{1}\)

Verify the identity \(\csc \theta \cos \theta \tan \theta=1\).

## Example \(\PageIndex{3A}\): Verifying the Equivalency Using the Even-Odd Identities

Verify the following equivalency using the even-odd identities:

\((1+\sin x)[1+\sin(−x)]={\cos}^2 x\)

Working on the left side of the equation, we have

\( (1+\sin x)[1+\sin(−x)]=(1+\sin x)(1-\sin x)\)

## Example \(\PageIndex{3B}\): Verifying a Trigonometric Identity Involving \({\sec}^2 \theta\)

Verify the identity \(\dfrac{{\sec}^2 \theta−1}{{\sec}^2 \theta}={\sin}^2 \theta\)

As the left side is more complicated, let’s begin there.

## Exercise \(\PageIndex{2}\)

Show that \(\dfrac{\cot \theta}{\csc \theta}=\cos \theta\).

## Example \(\PageIndex{4}\): Creating and Verifying an Identity

\(2 \tan \theta \sec \theta=\dfrac{2 \sin \theta}{1−{\sin}^2 \theta}\)

## Example \(\PageIndex{5}\): Verifying an Identity Using Algebra and Even/Odd Identities

\(\dfrac{{\sin}^2(−\theta)−{\cos}^2(−\theta)}{\sin(−\theta)−\cos(−\theta)}=\cos \theta−\sin \theta\)

Let’s start with the left side and simplify:

## Exercise \(\PageIndex{3}\)

## Example \(\PageIndex{6}\): Verifying an Identity Involving Cosines and Cotangents

Verify the identity: \((1−{\cos}^2 x)(1+{\cot}^2 x)=1\).

## Using Algebra to Simplify Trigonometric Expressions

## Example \(\PageIndex{7A}\): Writing the Trigonometric Expression as an Algebraic Expression

## Example \(\PageIndex{7B}\): Rewriting a Trigonometric Expression Using the Difference of Squares

Rewrite the trigonometric expression using the difference of squares: \(4{cos}^2 \theta−1\).

## Exercise \(\PageIndex{4}\)

Rewrite the trigonometric expression using the difference of squares: \(25−9{\sin}^2 \theta\).

This is a difference of squares formula: \(25−9{\sin}^2 \theta=(5−3\sin \theta)(5+3\sin \theta)\).

## Example \(\PageIndex{8}\): Simplify by Rewriting and Using Substitution

Simplify the expression by rewriting and using identities:

\({\csc}^2 \theta−{\cot}^2 \theta\)

We can start with the Pythagorean identity.

## Exercise \(\PageIndex{5}\)

(Hint: Multiply the numerator and denominator on the left side by \(1−\sin \theta\).)

## Key Equations

- There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to simplify a problem.
- Graphing both sides of an identity will verify it. See Example \(\PageIndex{1}\).
- Simplifying one side of the equation to equal the other side is another method for verifying an identity. See Example \(\PageIndex{2}\) and Example \(\PageIndex{3}\).
- The approach to verifying an identity depends on the nature of the identity. It is often useful to begin on the more complex side of the equation. See Example \(\PageIndex{4}\).
- We can create an identity and then verify it. See Example \(\PageIndex{5}\).
- Verifying an identity may involve algebra with the fundamental identities. See Example \(\PageIndex{6}\) and Example \(\PageIndex{7}\).
- Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques throughout this text, as they consist of the fundamental rules of mathematics. See Example \(\PageIndex{8}\), Example \(\PageIndex{9}\), and Example \(\PageIndex{10}\).

## Trigonometric Equations and Identities Solved Examples

BC / AB = (Opposite side) / hypotenuse, gives sine of A and denoted as sin A.

AC / AB = (adjacent side) / hypotenuse, gives the cosine of A and denoted as cos A.

BC / AC = (Opposite side) /(adjacent side), gives the tangent of A and denoted as tan A.

## Trigonometric Equations And Identities

Steps to solve a trigonometric equation are as follows:

## List of Trigonometric Equations And Identities

## General solution of some Trigonometric equations

c) cos θ = 0 ⇒ θ = (2n + 1) π/2

d) sin θ = 1 ⇒ θ = (4n + 1) π/2

e) cos θ = – 1 ⇒ θ = (4n – 1) π/2

## General Solution of some Standard Equations

a) sin θ = sin α ⇒ θ = nπ + (-1) n θ

b) cos θ = cos α ⇒ θ = 2nπ ± θ

Angle-Sum and Difference Identities

- sin (α + β) = sin (α)cos (β) + cos (α)sin (β)
- sin (α – β) = sin (α)cos (β) – cos (α)sin (β)
- cos (α + β) = cos (α)cos (β) – sin (α)sin (β)
- cos (α – β) = cos (α)cos (β) + sin (α)sin (β)
- tan (A + B) = (tan A + tan B)/(1 – tan A tan B)
- tan (A – B) = (tan A – tan B)/(1 + tan A tan B)
- cot (A + B) = (cot A cot B – 1)/(cot A + cot B)
- cot (A – B) = (cot A cot B + 1)/(cot B – cot A)

- sin (-A) = – sin A
- cos (-A) = cos A
- tan (-A) = – tan A
- cosec (-A) = – cosec A
- sec (-A) = sec A
- cot (-A) = – cot A

Trigonometric Equations and its Solutions

Trigonometry Previous Year Questions with Solutions

## Trigonometric Equations And Identities Examples

Example 1: In Δ ABC, sin (A − B) / sin (A + B) = ?

But cos B = [a 2 + c 2 − b 2 ] / 2ac,cos A = b 2 + c 2 − a 2 / 2bc

⇒ [a / c] [cos B] − [b / c] [cos A] = 1 / 2c 2

= (a 2 + c 2 − b 2 − b 2 – c 2 + a 2 ) / 2c 2

Hence, the angles are 18 o , 36 o , 126 o

Greatest side ∝ sin(126 o ) Smallest side ∝ sin(18 o ) and

Given expression is sin 2 5 o + sin 2 10 o + sin 2 15 o + . . . + sin 2 85 o + sin 2 90 o .

We know that sin 90 o = 1 or sin 2 90 o = 1.

Similarly, sin 45 o =1 / √2 or sin 2 45 o = 1 / 2 and the angles are in A.P. of 18 terms.

We also know that sin 2 85 o = [sin (90 o −5 o )] 2 = cos 2 5 o .

Therefore, from the complementary rule, we find sin 2 5 o + sin 2 85 o = sin 2 5 o +cos 2 5 o = 1

Example 4: Find the value of 1 + cos 56 o + cos58 o − cos66 o .

1 + cos 56 o + cos58 o − cos66 o

Apply the conditional identity

We get the value of the required expression equal to 4 cos28 o cos29 o sin33 o .

Adding and subtracting the given relation, we get

\(\begin{array}{l}(m-n)=a\,\,{{(\cos \alpha -\sin \alpha )}^{3}}\end{array} \)

We have tan 2x = 2 tan x /(1-tan 2 x)

So tan π/4 = 2 tan π/8 /(1-tan 2 π/8)

π/8 lies in the first quadrant. So tan π/8 is positive.

## Trigonometric Equations – Important Topics

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## Module 9: Trigonometric Identities and Equations

Solving trigonometric equations with identities, learning outcomes.

- Verify the fundamental trigonometric identities.
- Simplify trigonometric expressions using algebra and the identities.

## Verify the fundamental trigonometric identities

Prove: [latex]1+{\cot }^{2}\theta ={\csc }^{2}\theta [/latex]

Figure 2. Graph of [latex]y=\sin \theta[/latex]

Recall that an even function is one in which

Figure 3. Graph of [latex]y=\cos \theta[/latex]

- Since [latex]\sin \left(-\theta \right)=-\sin \theta[/latex], sine is an odd function.
- Since, [latex]\cos \left(-\theta \right)=\cos \theta[/latex], cosine is an even function.

## A General Note: Summarizing Trigonometric Identities

The Pythagorean identities are based on the properties of a right triangle.

The reciprocal identities define reciprocals of the trigonometric functions.

The quotient identities define the relationship among the trigonometric functions.

## Example 1: Graphing the Equations of an Identity

## Analysis of the Solution

## How To: Given a trigonometric identity, verify that it is true.

- Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to simplify than to build.
- Look for opportunities to factor expressions, square a binomial, or add fractions.
- Noting which functions are in the final expression, look for opportunities to use the identities and make the proper substitutions.
- If these steps do not yield the desired result, try converting all terms to sines and cosines.

## Example 2: Verifying a Trigonometric Identity

Verify [latex]\tan \theta \cos \theta =\sin \theta[/latex].

We will start on the left side, as it is the more complicated side:

Verify the identity [latex]\csc \theta \cos \theta \tan \theta =1[/latex].

## Example 3: Verifying the Equivalency Using the Even-Odd Identities

Verify the following equivalency using the even-odd identities:

[latex]\left(1+\sin x\right)\left[1+\sin \left(-x\right)\right]={\cos }^{2}x[/latex]

Working on the left side of the equation, we have

## Example 4: Verifying a Trigonometric Identity Involving sec 2 θ

Verify the identity [latex]\frac{{\sec }^{2}\theta -1}{{\sec }^{2}\theta }={\sin }^{2}\theta[/latex]

As the left side is more complicated, let’s begin there.

Show that [latex]\frac{\cot \theta }{\csc \theta }=\cos \theta[/latex].

## Example 5: Creating and Verifying an Identity

[latex]2\tan \theta \sec \theta =\frac{2\sin \theta }{1-{\sin }^{2}\theta }[/latex]

## Example 6: Verifying an Identity Using Algebra and Even/Odd Identities

Let’s start with the left side and simplify:

## Example 7: Verifying an Identity Involving Cosines and Cotangents

Verify the identity: [latex]\left(1-{\cos }^{2}x\right)\left(1+{\cot }^{2}x\right)=1[/latex].

We will work on the left side of the equation.

## Simplify trigonometric expressions using algebra and the identities

## Example 8: Writing the Trigonometric Expression as an Algebraic Expression

## Example 9: Rewriting a Trigonometric Expression Using the Difference of Squares

Rewrite the trigonometric expression: [latex]4{\cos }^{2}\theta -1[/latex].

Rewrite the trigonometric expression: [latex]25 - 9{\sin }^{2}\theta [/latex].

## Example 10: Simplify by Rewriting and Using Substitution

Simplify the expression by rewriting and using identities:

[latex]{\csc }^{2}\theta -{\cot }^{2}\theta [/latex]

We can start with the Pythagorean identity.

[latex]1+{\cot }^{2}\theta ={\csc }^{2}\theta [/latex]

(Hint: Multiply the numerator and denominator on the left side by [latex]1-\sin\theta[/latex]).

## Key Equations

- There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to simplify a problem.
- Graphing both sides of an identity will verify it.
- Simplifying one side of the equation to equal the other side is another method for verifying an identity.
- The approach to verifying an identity depends on the nature of the identity. It is often useful to begin on the more complex side of the equation.
- We can create an identity by simplifying an expression and then verifying it.
- Verifying an identity may involve algebra with the fundamental identities.
- Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques throughout this text, as they consist of the fundamental rules of mathematics.

## IMAGES

## VIDEO

## COMMENTS

Identities enable us to simplify complicated expressions. They are the basic tools of trigonometry used in solving trigonometric equations

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Problems on Trigonometric Identities ; 1. (1 - sin A)/(1 + sin A) = (sec A - tan A) ; 2. Prove that, √{(sec θ – 1)/(sec θ + 1)} = cosec θ - cot θ. ; 3. tan4 θ +

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this Video contains plenty of example and practice problems on how to prove trigonometric identitiesif you like this video just like and

Trigonometric Equations And Identities Examples ; We have tan 2x = 2 tan x /(1-tan · x) ; So tan π/4 = 2 tan π/8 /(1-tan · π/8) ; ∴ 1 = 2y / (1-y · ) ; ⇒ 1-y · = 2y.

The other even-odd identities follow from the even and odd nature of the sine and cosine functions. For example, consider the tangent identity, tan(

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