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# 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

Note by Yan Yau Cheng
3 years, 8 months ago