Question

Tutorial: Trigonometric Square Identities

Answer 1

This is a fun approach to understanding the square identities, nothing special. :)

Most people are familiar with this technique:
We start with our square identity involving sine and cosine,\[\large\rm \color{blue}{\sin^2\theta+\cos^2\theta=1}\]Dividing both sides by \(\large\rm\sin^2\theta\) leads to our next square identity,\[\large\rm \frac{\sin^2\theta}{\sin^2\theta}+\frac{\cos^2\theta}{\sin^2\theta}=\frac{1}{\sin^2\theta}\quad\to\quad \color{red}{1+\cot^2\theta=\csc^2\theta}\]Instead, dividing by \(\Large\rm \cos^2\theta\) leads to our third square identity,\[\large\rm \frac{\sin^2\theta}{\cos^2\theta}+\frac{\cos^2\theta}{\cos^2\theta}=\frac{1}{\cos^2\theta}\quad\to\quad \color{limegreen}{\tan^2\theta+1=\sec^2\theta}\]This is certainly a fine way to generate the three square identities.


But there's another interesting way I want to show you,
using triangles and the unit circle,
just in case you haven't seen it before.

Give me a couple minutes to draw the diagram please. :)

Answer 2

We draw our angle, \(\Large\rm \theta\), in the first quadrant arbitrarily.

Answer 3

Since this is the unit circle, the radius is always 1.
Notice that the radius is in a different location for each triangle!

Answer 4

For our blue triangle, it's the hypotenuse.