Identities of Inverse Operations

The easiest first step is to simplify our inverse identities:

Cross cancelling, we end up with

Finally, eliminate the fraction:

Thus,

Using the quotient identities for trig functions, you can rewrite,

and

Then the fraction becomes

Use the Pythagorean Identities:

and

Thus the expression becomes,

.

Use the distributive property (FOIL method) to simplify the expression.

Using Pythagorean Identities:

.

First, simplify the first term in the expression to 1 because of the Pythagorean Identity.

Then, simplify the second term to

.

This reduces to

.

The expression is now,

.

Distribute the negative and get,

.

Divide both sides by to get .

Take the square root of both sides to get that and .

The angles for which this is true (this is taking the arctan) are every angle when and .

These angles are all the multiples of .

Let's look at these individually:

is true by definition, as is .

is also true because of a co-function identity.

This leaves two - and we can tell which of these does not work using the fact that , which means that is our answer.