Trigonometric Identities - Pythagorean Identities

In addition to the basic trig identities that I've already covered, a little bit of derivation leads to a whole bunch of new identities. In addition to the Pythagorean Trigonometric Identity I covered in my last post, there are a few other identities that can be derived from that (also sometimes called Pythagorean Identities).

If we start with the first identity:

we can divide each side by the Sine term to give something new.

[Sin^2(theta) + C! os^2(theta)] / Sin^2(theta) = 1 / Sin^2(theta).

The first term reduces to 1:
1 + Cos^2(theta) / Sin^2(theta) = 1 / Sin^2(theta)

And the remaining terms can be simplified and rewritten in terms of inverse functions:
1 + [Cos(theta) / Sin(theta)]^2 = [1 / Sin(theta)]^2
1 + Cot^2(theta) = Csc^2(theta).

More likely, it will look something like this:



Similarly, using the same steps as above (which I will leave for you to play with if you want), you can start with the first identity, and divide by the Cosine term (rather than the Sine term), to come up with the third Pythagorean Trigonometric Identity:
1 + tan^2(theta) = sec^2(theta)




There are still several other Trig identities which I will show you how to derive in coming posts. Stay tuned!

trigonometric identities