Using tan (A+B), find the exact value of tan (105).

-2-√3

Step-by-step explanation:

The sum of angles formula for tangent is ...

tan (A+B) = (tan (A) + tan (B)) / (1 - tan (A) tan (B))

For the desired angle, we can use A=45°, B=60°. Then we have ...

tan (105°) = (tan (45°) + tan (60°)) / (1 - tan (45°) tan (60°))

The tangent values we need are ...

tan (45°) = 1

tan (60°) = √3

Then our tangent is ...

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

tan (105°) = - (2+√3)