Trigonometry: Basic Trig Identities

This post is part of a series of posts on Trigonometry. To see all the posts, click on the tag #TrigonometryTutorials below. This is the post you should read before you read this.

Here are a few trig identities that are very important to remember:

Identity 1: $\sin^2 x+ \cos^2 x = 1$

Proof: Use the Definition of the trig functions: $\sin x = \frac oh$ and $\cos \frac ah$

$\left(\frac oh\right)^2+\left(\frac ah\right)^2= 1$ $\frac {o^2}{h^2} + \frac {a^2}{h^2} = 1$

Multiply both sides by $h^2$

$a^2+o^2=h^2$

This is just the Pythagorean Theorem! So this holds true. $\blacksquare$

A variation of Identity 1 can be achieved by:

dividing both sides by $\sin^2 x$

$1+ \cot^2 x = \csc x$

dividing both sides by $\cos^2 x$

$\tan^2 x +1 = \sec x$

Identity 2: $\sin x = \cos (90^\circ-x)$

In a Right angled triangle, if one of the angles is $x$, The other angle will be $90-x$ (the angle other than the right angle, that is.). The opposite angle of one angle is the adjacent of the other angle, therefore $\sin x = \cos (90^\circ-x)$.

Using the same reasoning, these other identities can be obtained:

$\cos x = \sin (90-x)$ $\sec x = \csc (90-x)$ $\csc x = \sec (90-x)$ $\tan x = \cot(90-x)$ $\cot x = \tan (90-x)$

The Next post in this series is here