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# Precalculus - Trigonometric Identities

## Introduction

• We know that equations involving trigonometric functions of a variable (angle) are termed Trigonometric equations.
• Trigonometric Identities are special trigonometric equations that hold true for all the values of the angles(s) involved.
• For example, $2\mathrm{sin}\theta -1=0$ is a trigonometric equation and not an identity, as it doesn't hold true for all the values of $\theta$. It is only true for some specific values of $\theta$.

## Trigonometric Identities

Trigonometric Identities can be divided into several groups, as mentioned below:

Reciprocal Identities

•   or,
•   or,
•   or,

Quotient Identities

Pythagorean Identities

Sign Identities / Opposite Angle Identities

Cofunction Identities

Double Angle Identities

Triple Angle Identities

Sum and Difference of Angle Identities

## Solved Examples

Example 1: Find the value of .

Solution:

[since, $1-{\mathrm{cos}}^{2}\theta ={\mathrm{sin}}^{2}\theta$]

Example 2: Find the value of .

Solution:

[since, $\mathrm{cos}e{c}^{2}\theta =1+\mathrm{co}{t}^{2}\theta$ $⇒\mathrm{co}{t}^{2}\theta =\mathrm{cos}e{c}^{2}\theta -1$]

[since $\mathrm{cos}ec\theta =\frac{1}{\mathrm{sin}\theta }$]

$=1$

Example 3: Find the value of $\mathrm{cos}2\theta -2{\mathrm{cos}}^{2}\theta +1$.

Solution: $\mathrm{cos}2\theta -2{\mathrm{cos}}^{2}\theta +1$

$=\left(2{\mathrm{cos}}^{2}\theta -1\right)-2{\mathrm{cos}}^{2}\theta +1$ [since, $\mathrm{cos}2\theta =2{\mathrm{cos}}^{2}\theta -1$]

$=2{\mathrm{cos}}^{2}\theta -1-2{\mathrm{cos}}^{2}\theta +1$

$=0$

Example 4: Find the value of .

Solution:

[since $se{c}^{2}\theta =1+{\mathrm{tan}}^{2}\theta$ ]

[since, $sec\theta =\frac{1}{\mathrm{cos}\theta }$]

$=1$

Example 5: Verify the identity .

Solution:

Simplifying LHS

[since, $\mathrm{cos}e{c}^{2}\theta =1+co{t}^{2}\theta$]

[since, $\mathrm{cos}ec\theta =\frac{1}{\mathrm{sin}\theta }$ and $sec\theta =\frac{1}{\mathrm{cos}\theta }$]

[since, $cot\theta =\frac{\mathrm{cos}\theta }{\mathrm{sin}\theta }$]

[since, $cot\theta =\frac{1}{\mathrm{tan}\theta }$]

$=cot\theta$

$=RHS$

•   or
•   or
•   or
•  and

## Blunder Areas

• All the trigonometric identities are trigonometric equations but not all trigonometric equations are trigonometric identities.