First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, can be split into .

Use the sum formula for tangent to simplify the expression. The formula states that .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

The exact value of is .

The exact value of is .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

The exact value of is .

Multiply .

Multiply by .

Multiply by .

The exact value of is .

Multiply by .

Multiply by .

Multiply and .

Expand the denominator using the FOIL method.

Simplify.

Reorder terms.

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify each term.

Multiply by .

Multiply by .

Multiply by .

Multiply .

Multiply by .

Multiply by .

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

Rewrite as .

Apply the power rule and multiply exponents, .

Combine and .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Evaluate the exponent.

Add and .

Subtract from .

Factor out of .

Factor out of .

Factor out of .

Move the negative one from the denominator of .

Rewrite as .

Apply the distributive property.

Multiply by .

Multiply by .

Multiply by .

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

Expand Using Sum/Difference Formulas tan(165)