How do you verify (1+tan^2x)/(1+cot^2x) = (tan^2x)^2?

1 Answer
Jul 20, 2016

These are called the Trigonometric Pythagorean Identities

Explanation:

Suppose you have a right triangle, with sides a and b, and hypotenuse c. Now, suppose that a is the side adjacent to the working angle. Then, by Pythagorean Theorem, you have a^2+b^2=c^2.

1+tan^2x=sec^2x
1+cot^2x=csc^2x

The first identity comes from dividing the equation by a^2, and the second by b^2. And remembering the basic trigonometric identities: Soh, Cah, Toa. There is a third identity, but I will leave that as an exercise for you, if you are interested.

Now, with those identities, we can substitute in the original expression:
(1+tan^2x)/(1+cot^2x)=(sec^2x)/(csc^2x)=(sin^2x)/(cos^2x)=(tan^2x)^2

I believe the last three parts of the equalities are well known facts, or at least extremely easy to show, hence I'll stop here with the proof.