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# Trigonometric Identities

Top
 Sub Topics An identity is an equation which is always true. For example, pythagorean theorem's (Base)$^{2}$ + (Perpendicular)$^{2}$ = (Hypotenuse)$^{2}$ is true for right triangles. Trigonometric identities are important identities that involve sums or differences of angles. Equalities that include the trigonometric functions are known as trigonometric identities.

## Reciprocal Identities

There are six basic trigonometric functions and their reciprocals are given below.
1. Sin x = $\frac{1}{\csc x}$
2. Cos x = $\frac{1}{\sec x}$
3. Tan x = $\frac{1}{\cot x}$
4. cot x = $\frac{1}{\tan x}$

## Pythagorean Identities

Given below are the Pythagorean identities.
1. Sin$^{2}$ x + Cos$^{2}$ x = 1
2. 1 + Tan$^{2}$ x = Sec$^{2}$ x
3. 1 + Cot$^{2}$ x = Csc$^{2}$ x

## Quotient Identities

There are two quotient identities to tell us that the tangent and cotangent functions can be expressed in terms of the sine and cosine functions. Given below are the quotient identities.

tan x = $\frac{\sin x}{\cos x}$

cot x = $\frac{\cos x}{\sin x}$

## Cofunction Identities

Below are some co-function identities:

Sin($\frac{\pi}{2}$ - x) = Cos x

Cos($\frac{\pi}{2}$ - x) = Sin x

Tan($\frac{\pi}{2}$ - x) = Cot x

Csc($\frac{\pi}{2}$ - x) = Sec x

Sec($\frac{\pi}{2}$ - x) = Csc x

Cot($\frac{\pi}{2}$ - x) = Tan x

## Even-Odd Identities

A function f(x) is said to be even if f(-x) = f(x) and a function f(x) is said to be odd if f(-x) = - f(x). Even-odd identities for Sin, Cos, Tan, Csc, Sec and Cot are given below.
1. Sin (-x) = - Sin x
2. Cos (-x) = Cos x
3. Tan (-x) = - Tan x
4. Csc (-x) = - Csc x
5. Sec (-x) = Sec x
6. Cot (-x) = - Cot x

## Sum-Difference Formulas

For the basic trigonometric functions, sum difference formula for Sin, Cos and Tan are given below:

Sin (x + y ) = Sin x Cos y + Cos x Sin y

Sin (x - y) = Sin x Cos y - Cos x Sin y

Cos(x + y ) = Cos x Cos y - Sin x Sin y

Cos(x - y ) = Cos x Cos y + Sin x Sin y

Tan (x + y) = $\frac{\tan x + \tan y}{1 - \tan x \times \tan y}$

Tan (x - y) = $\frac{\tan x - \tan y}{1 + \tan x \times \tan y}$

## Double Angle Formula

For the basic trigonometric functions, given below are the double angle formulas for Sin, Cos and Tan.

Sin 2x = 2 Sinx Cosx
Cos 2x = $\begin{Bmatrix} \cos^{2}x - \sin^{2}x\\ 2 \cos^{2}x -1\\ 1 - 2 \sin^{2}x\end{Bmatrix}$
Tan 2x = $\frac{2 \tan x}{1 - \tan ^{2} x}$

## Half Angle Formula

Given below are the half angle formula's for basic trigonometric functions.

Sin ($\frac{x}{2}$) = $\pm$ $\sqrt{\frac{1 - \cos x}{2}}$
Cos($\frac{x}{2}$)= $\pm$ $\sqrt{\frac{1 + \cos x}{2}}$
Tan($\frac{x}{2}$)= $\left\{\begin{matrix} \pm\sqrt{\frac{1 - \cos x}{1 + \cos x}}\\ \frac{\sin x}{1 + \cos x}\\ \frac{1 - \cos x}{\sin x}\end{matrix}\right.$

## Sum-to-Product Formulas

Sum to product formula for Sin and Cos are given below:

1. Sin A + Sin B = 2 Sin $\frac{A+B}{2}$ . Cos $\frac{A-B}{2}$

2. Sin A - Sin B = 2 Cos $\frac{A+B}{2}$ . Sin $\frac{A-B}{2}$

3. Cos A + Cos B = 2 Cos $\frac{A+B}{2}$ . Cos $\frac{A-B}{2}$

4. Cos A - Cos B = - 2 Sin $\frac{A+B}{2}$ . Sin $\frac{A-B}{2}$

## Product-to-Sum Formulas

Product to sum formula for Sin and Cos are given below:

1. Sin A . Cos B = $\frac{1}{2}$ [ Sin ( A + B ) + Sin ( A - B ) ]

2. Cos A. Sin B = $\frac{1}{2}$ [ Sin ( A + B ) - Sin ( A - B ) ]

3. Cos A. Cos B = $\frac{1}{2}$ [ Cos ( A + B ) + Cos ( A - B ) ]

4. Sin A . Sin B = - $\frac{1}{2}$ [ Cos ( A + B ) - Cos ( A - B ) ]
= $\frac{1}{2}$ [ Cos ( A - B ) - Cos ( A + B ) ]

## Trigonometric Identities Problems

Given below are some of the problems on trigonometric identities.

### Solved Example

Question: If Tan $\theta$ = 5, find Tan 3$\theta$
Solution:
We know the formula for Tan (3$\theta$)
Tan(3$\theta$) = $\frac{3 \tan \theta - \tan^{3} \theta}{1 - 3 \tan^{2} \theta}$

Plugging in Tan $\theta$ = 5 in the above identity, we get

Tan (3$\theta$) = $\frac{3(5) - 5^{3}}{1-3(5)^{2}}$

= $\frac{15-125}{-74}$

= $\frac{-110}{-74}$

= 1.49

### Practice Problem

Question: Check whether the identity $\frac{(\sec^{2} \theta - 1 )}{(\sec^{2} \theta)}$ = Sin$^{2}\theta$ is true.