Two different versions of the Trigonometric Identity using the Pythagorean Theorem.

In my class, I call

sin²alpha + cos²alpha= 1

The Trigonometric Identity.

Other sources say there are many trigonometric identities, but the vast majority of them are just re-workings of this formula.  Here are two examples that look different but are very closely related.

Consider the following lengths of the sides for a right triangle

a = sinalpha, b = cosalpha, c = 1

These are the side lengths in what I call the Normal Standard Position Right Triangle, where the right angle is the lower right and alpha is the lower left and the hypotenuse has length 1.

What if we have a similar triangle, but instead of the hypotenuse having length 1, side b was length 1?  We can do this by dividing all the side lengths by cosalpha.

a = sinalpha/cosalpha, b = cosalpha/cosalpha, c = 1/cosalpha

or

a = tanalpha, b = 1, c = secalpha

Since this is still a right triangle, we get a new version of the Pythagorean Theorem

tan²alpha + 1 = sec²alpha

often re-written as


tan²alpha = sec²alpha - 1


The other possibility is to have a have length 1.  We do this by dividing all length by sinalpha.

a = sinalpha/sinalpha, b = cosalpha/sinalpha, c = 1/sinalpha

or

a = 1, b = cotalpha, c = cscalpha

This gives us yet another version of the Trigonometric Identity


cot²alpha + 1 = csc²alpha


often re-written as

cot²alpha = csc²alpha - 1



This brings us to the Complementary Identity rule.  If you have any trig formula, you can replace every trig function with its complement and you have another trig formula.  For example,


tanalpha = sinalpha/cosalpha

Change tan to cot, sin to cos and cos to sin and we get



cotalpha = cosalpha/sinalpha