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, $2\sin \theta -1=0$ 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

• $\sin \theta =\frac{1}{\cos ec \theta }$  or,  $\cos ec \theta =\frac{1}{\sin \theta }$
• $\cos \theta =\frac{1}{sec \theta }$  or,  $sec \theta =\frac{1}{\cos \theta }$
• $\tan \theta =\frac{1}{cot \theta }$  or,  $cot \theta =\frac{1}{\tan \theta }$

 

Quotient Identities

• $\tan \theta =\frac{\sin \theta }{\cos \theta }$
• $cot \theta =\frac{\cos \theta }{sin \theta }$

 

Pythagorean Identities

• ${\sin }^2 \theta +{\cos }^2 \theta =1$
• $se{c}^2 \theta =1+{\tan }^2 \theta$
• $cose{c}^2 \theta =1+co{t}^2 \theta$

 

Sign Identities / Opposite Angle Identities

• $\sin \left(-\theta \right)=-\sin \theta$
• $\cos \left(-\theta \right)=\cos \theta$
• $\tan \left(-\theta \right)=-\tan \theta$

 

Cofunction Identities

• $\sin \left(\frac{\mathrm{\pi }}{2}-\theta \right)=\cos \theta$
• $\cos \left(\frac{\mathrm{\pi }}{2}-\theta \right)=\sin \theta$
• $\tan \left(\frac{\mathrm{\pi }}{2}-\theta \right)=cot \theta$
• $cot\left(\frac{\mathrm{\pi }}{2}-\theta \right)=\tan \theta$
• $sec\left(\frac{\mathrm{\pi }}{2}-\theta \right)=\cos ec \theta$
• $\cos ec\left(\frac{\mathrm{\pi }}{2}-\theta \right)=sec \theta$

 

Double Angle Identities

• $\sin 2\theta =2\sin \theta ·\cos \theta$$=\frac{2\tan \theta }{1+{\tan }^2 \theta }$
• $\cos 2\theta ={\cos }^2 \theta -si{n}^2 \theta$$=2{\cos }^2 \theta -1$$=1-2si{n}^2 \theta$
• $\tan 2\theta =\frac{2\tan \theta }{1-\tan { }^2\theta }$

 

Triple Angle Identities

• $\sin 3\theta =3\sin \theta -4{\sin }^3 \theta$
• $\cos 3\theta =4{\cos }^3 \theta -3\cos \theta$

 

Sum and Difference of Angle Identities

• $\sin x+\sin y=2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$
• $\sin x-\sin y=2\cos \left(\frac{x+y}{2}\right)\sin \left(\frac{x-y}{2}\right)$
• $\cos x+\cos y=2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)$
• $\cos x-\cos y=2\sin \left(\frac{x+y}{2}\right)\sin \left(\frac{y-x}{2}\right)$

Solved Examples

Example 1: Find the value of $\left(1-{\cos }^2\theta \right) \cos e{c}^2\theta$.

Solution: $\left(1-{\cos }^2\theta \right) \cos e{c}^2\theta$

$=\left({\sin }^2\theta \right) \cos e{c}^2\theta$  [since, $1-{\cos }^2\theta ={\sin }^2\theta$]

$=\left(\frac{1}{\mathrm{co}se{c}^2\theta }\right) \cos e{c}^2\theta = 1$

Example 2: Find the value of $\sin \theta +\frac{\sin \theta }{{\tan }^2\theta \left(1+\cos ec\theta \right)}$.

Solution: $\sin \theta +\frac{\sin \theta }{{\tan }^2\theta \left(1+\cos ec\theta \right)}$

$=\sin \theta \left[1+\frac{1}{{\tan }^2\theta \left(1+\cos ec\theta \right)}\right]$

$=\sin \theta \left[1+\frac{co{t}^2\theta }{ \left(1+\cos ec\theta \right)}\right]$

$=\sin \theta \left[1+\frac{\left(\cos e{c}^2\theta -1\right)}{ \left(1+\cos ec\theta \right)}\right]$  [since, $\cos e{c}^2\theta =1+\mathrm{co}{t}^2\theta$ $\Rightarrow \mathrm{co}{t}^2\theta =\cos e{c}^2\theta -1$]

$=\sin \theta \left[1+\frac{\left(\cos ec\theta -1\right) \left(\cos ec\theta -1\right)}{ \left(1+\cos ec\theta \right)}\right]$

$=\sin \theta \left[1+\left(\cos ec\theta -1\right)\right]$

$=\sin \theta \left[1+\cos ec\theta -1\right]$

$=\sin \theta \left[\cos ec\theta \right]$

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

$=1$

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

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

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

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

$=0$

Example 4: Find the value of $\left(1-\sin \theta \right) \left(1+\sin \theta \right) \left(1+{\tan }^2\theta \right)$.

Solution: $\left(1-\sin \theta \right) \left(1+\sin \theta \right) \left(1+{\tan }^2\theta \right)$

$=\left(1-{\sin }^2\theta \right) \left(se{c}^2\theta \right)$ [since $se{c}^2\theta =1+{\tan }^2\theta$ ]

$=\left(co{s}^2\theta \right) \left(\frac{1}{co{s}^2\theta }\right)$ [since, $sec\theta =\frac{1}{\cos \theta }$]

$=1$

Example 5: Verify the identity $\frac{\left(1+\mathrm{co}{t}^2\theta \right) \tan \theta }{se{c}^2\theta }=cot\theta$.

Solution: $\frac{\left(1+\mathrm{co}{t}^2\theta \right) \tan \theta }{se{c}^2\theta }=cot\theta$

Simplifying LHS

$\frac{\left(1+\mathrm{co}{t}^2\theta \right) \tan \theta }{se{c}^2\theta }$

$=\frac{\left(\cos e{c}^2\theta \right) \tan \theta }{se{c}^2\theta }$ [since, $\cos e{c}^2\theta =1+co{t}^2\theta$]

$=\frac{{\cos }^2\theta \tan \theta }{{\sin }^2\theta }$ [since, $\cos ec\theta =\frac{1}{\sin \theta }$ and $sec\theta =\frac{1}{\cos \theta }$]

$=\mathrm{co}{t}^2\theta \tan \theta$ [since, $cot\theta =\frac{\cos \theta }{\sin \theta }$]

$=\frac{\mathrm{co}{t}^2\theta }{cot\theta }$ [since, $cot\theta =\frac{1}{\tan \theta }$]

$=cot\theta$

$=RHS$

Cheat Sheet

• $\sin \theta =\frac{1}{\cos ec \theta }$  or  $\cos ec \theta =\frac{1}{\sin \theta }$
• $\cos \theta =\frac{1}{sec \theta }$  or  $sec \theta =\frac{1}{\cos \theta }$
• $\tan \theta =\frac{1}{\mathrm{co}t \theta }$  or  $cot \theta =\frac{1}{\tan \theta }$
• $\tan \theta =\frac{\sin \theta }{\cos \theta }$ and $cot \theta =\frac{\cos \theta }{sin \theta }$
• ${\sin }^2 \theta +co{s}^2 \theta =1$
• $se{c}^2 \theta =1+{\tan }^2 \theta$
• $cose{c}^2 \theta =1+co{t}^2 \theta$
• $\sin \left(-\theta \right)=-\sin \theta$
• $cos\left(-\theta \right)=cos \theta$
• $\tan \left(-\theta \right)=-\tan \theta$
• $\sin 2\theta =2\sin \theta ·\cos \theta$
• $\cos 2\theta ={\cos }^2 \theta -{\sin }^2 \theta =2{\cos }^2 \theta -1=1-2{\sin }^2 \theta$
• $\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2 \theta }$
• $\sin \left(\frac{\mathrm{\pi }}{2}-\theta \right)=\cos \theta$
• $cos\left(\frac{\mathrm{\pi }}{2}-\theta \right)=\sin \theta$
• $\mathrm{ta}n\left(\frac{\mathrm{\pi }}{2}-\theta \right)=\mathrm{co}t \theta$

Blunder Areas

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