simplify each expression. your final answer should be exact and should not contain any trigonometric expression

Question

1. cos(tan⁻¹(2))
2. sin(cos⁻¹(x))

Answer 1

1) cos(tan^-1(2)) = 1/√1 + tan²(tan^-1(2)) = 1/√1 + 2² = 1/√5

2) sin(cos^-1(x)) = √1 - cos²(cos^-1(x)) = √1 - x²

Answer 2

Let θ = Tan^-12. Then Tanθ = 2 = 2/1. Draw a right triangle with θ as one of the acute angles. The side opposite θ has length 2 and the side adjacent to θ has length 1. So, by the Pythagorean Theorem, the length of the hypotenuse is √5.

So, cos(Tan^-12) = cosθ = 1/√5

Do the other one in a similar manner letting θ = cos^-1x.