Integration with trigonometric identities

In a nutshell

Many integrals can be simplified by using any number of trigonometric identities.

List of trigonometric identities

Here is a list of all the trigonometric identities you need to know.

description	identity
Basic definitions	$\tan(x)\equiv \dfrac{\sin(x)}{\cos(x)}$
	$\cosec(x)\equiv \dfrac{1}{\sin(x)}$	$\sec(x) \equiv \dfrac{1}{\cos(x)}$	$\cot(x)\equiv \dfrac{1}{\tan(x)}\equiv \dfrac{\cos(x)}{\sin(x)}$
Basic identities	$\sin^2(x)+\cos^2(x)\equiv 1$
	$\sec^2(x)\equiv 1 + \tan^2(x)$
	$\cosec^2(x) \equiv 1+ \cot^2(x)$
Compound angle identities	$\sin(A+B)\equiv \sin(A)\cos(B)\pm \cos(A)\sin(B)$
	$\cos(A\pm B)\equiv \cos(A)\cos(B)\mp\sin(A)\sin(B)$
	$\tan(A+B)=\dfrac{\tan(A)\pm \tan(B)}{1\mp \tan(A)\tan(B)}$
Double angle identities	$\sin(2x)\equiv 2\sin(x)\cos(x)$
	$\begin{aligned}\cos(2x)&\equiv \cos^2(x)- \sin^2(x)\\&\equiv2\cos^2(x)-1\\ &\equiv 1-2\sin^2(x)\end{aligned}$
	$\tan(2x)\equiv \dfrac{2\tan(x)}{1-\tan^2(x)}$

These can be used to make certain integrals much simpler.

Example 1

Compute the following integrals:

i) $\int \cos^2(x)\,dx$ 

ii) $\int \dfrac{1}{\cosec(2x)}\left(\dfrac{\cos(x)}{\sin(x)}+\dfrac{\sin(x)}{\cos(x)}\right)\, dx$ 

Part i):

Use the double angle formula to express $\cos^2(x)$ in terms of $\cos(2x)$ :

$\begin{aligned} \cos(2x)&\equiv 2\cos^2(x)-1\\ \Rightarrow \cos^2(x)&\equiv \dfrac{\cos(2x)+1}{2} \end{aligned}$

This is easier to integrate:

$\begin{aligned} \int \cos^2(x) \, dx&\equiv \int \left(\dfrac{\cos(2x)+1}{2}\right) \, dx\\&= \dfrac12\int \cos(2x)\, dx + \int\dfrac12 \, dx\\&=\dfrac12 \left(\dfrac12 \sin(2x)\right) + \dfrac12 x+C\\&=\dfrac14 \sin(2x) + \dfrac12x+C \end{aligned}$

$\int \cos^2(x) =\underline{\dfrac14 \sin(2x) + \dfrac12x+C}$

Part ii):

Simplify the integrand by writing everything as one fraction:

$\int \dfrac{1}{\cosec(2x)}\left(\dfrac{\cos(x)}{\sin(x)}+\dfrac{\sin(x)}{\cos(x)}\right)\, dx \equiv \int\dfrac{1}{\cosec(2x)}\left( \dfrac{\cos^2(x)+\sin^2(x)}{\sin(x)\cos(x)}\right)\,dx$

Use the following identities to simplify this:

$\cos^2(x)+\sin^2(x) \equiv 1\\ \sin(2x)\equiv 2\sin(x)\cos(x) \Rightarrow \sin(x)\cos(x) \equiv \dfrac12 \sin(2x)\\ \cosec(x)\equiv \dfrac{1}{\sin(x)} \Rightarrow \cosec(2x) \equiv \dfrac{1}{\sin(2x)}$

The integral becomes:

$\begin{aligned} \int\dfrac{1}{\cosec(2x)}\left( \dfrac{\cos^2(x)+\sin^2(x)}{\sin(x)\cos(x)}\right)\,dx &=\int \dfrac{1}{\frac{1}{\sin(2x)}} \left(\dfrac{1}{\frac12 \sin(2x)}\right) \, dx\\&=\int \sin(2x)\times \dfrac{2}{\sin(2x)}\, dx\\ &=\int 2\, dx\\&=2x+C \end{aligned}$

$\int \dfrac{1}{\cosec(2x)}\left(\dfrac{\cos(x)}{\sin(x)}+\dfrac{\sin(x)}{\cos(x)}\right)\, dx=\underline{2x+C}$