# Derivative Proofs of Inverse Trigonometric Functions

To prove these derivatives, we need to know pythagorean identities for trig functions.

Proving **arcsin(x)** (or **sin ^{-1}(x)**) will be a good example for being able to prove the rest.

## Derivative Proof of arcsin(x)

Prove

We know that

Taking the derivative of both sides, we get

We divide by cos(y)

Using a pythagorean identity for trig functions

*pythagorean identity*

We can substitute for cos(y)

Then we can substitute **sin ^{-1}(x)** back in for

**y**and

**x**for

**sin(y)**

There you have it! The best part is, the other inverse trig proofs are proved similarly by using pythagorean identities and substitution, except the cofunctions will be negative.

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