Question

Use a sum or difference identity to find the exact value of the expression.

tan 105° tan
105° =
(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Answer 1

To calculate tan 105°, use the sum identity for tangent with values 60° and 45°. Substitute the known tangent values for 60° (√3) and 45° (1) and simplify using the conjugate, resulting in the exact value of -2 - √3.

Step-by-step explanation:

To find the exact value of tan 105°, we can use the sum identity for tangent, which says that tan(A + B) = (tan A + tan B) / (1 - tan A * tan B). Knowing that 105° can be expressed as 60° + 45°, we set A to 60° and B to 45°.

Using the exact values for tan 60° (√3) and tan 45° (1), we substitute these into the identity:

tan 105° = (tan 60° + tan 45°) / (1 - tan 60° * tan 45°)
tan 105° = (√3 + 1) / (1 - √3 * 1)
tan 105° = (√3 + 1) / (1 - √3).

To simplify this expression, we can multiply the numerator and the denominator by the conjugate of the denominator, (1 + √3). This eliminates the radical in the denominator:

tan 105° = ((√3 + 1)(1 + √3)) / ((1 - √3)(1 + √3))
tan 105° = (√3^2 + 2√3 + 1) / (1 - √3^2)
tan 105° = (3 + 2√3 + 1) / (1 - 3)
tan 105° = (4 + 2√3) / (-2)
tan 105° = -2 - √3.

So the exact value of tan 105° is -2 - √3.