Split into two angles where the values of the six trigonometric functions are known.

Apply the sum of angles identity.

The exact value of is .

The exact value of is .

The exact value of is .

The exact value of is .

Simplify the denominator.

Multiply by .

Rewrite as .

Multiply by .

Multiply and .

Expand the denominator using the FOIL method.

Simplify.

Simplify the numerator.

Raise to the power of .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Rewrite as .

Expand using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify and combine like terms.

Simplify each term.

Multiply by .

Multiply by .

Multiply by .

Combine using the product rule for radicals.

Multiply by .

Rewrite as .

Pull terms out from under the radical, assuming positive real numbers.

Add and .

Add and .

Cancel the common factor of and .

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 .

The result can be shown in multiple forms.

Exact Form:

Decimal Form:

Find the Exact Value tan((7pi)/12)