Evaluate the expression without using a calculator:

Question

cos(tan^-1(-2))

Answer 1

cos[tan^{- 1}(- 2)].

Let tan ^{- 1}(- 2) = x.

Then, tan x = - 2.

• tan x = opposite/adjacent = - 2/1.

Phythagoras theorem : hypotenuse² = adjacent² + opposite² .

hypotenuse² = (- 2)² + 1²

hypotenuse² = 4 + 1 = 5

hypotenuse = √5.

⇒ cos x = adjacent/hypotenuse = 1/√5.

Therefore, cos[tan^{- 1}(- 2) = 1/√5.