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Trigonometric Identities

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.