Question

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

Answer 1

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.