How would you write cos(tan^-1u) as an algebraic expression in u?

Question

I know the answer, and I've seen tutorial on how to do it using pathagorus, but how would you do this using identities?

Answer 1

draw the triangle

In the triangle, label the sides 1 and u, and the hypotenuse √(1+u^2)

the angle tan^-1(u) is opposite the "u" side, since opposite/adjacent = u/1 = u

So, the cosine of that angle is adjacent/hypotenuse = 1/√(1+u^2)

You can also note that

cosθ = 1/secθ
sec^2θ = 1+tan^2θ, so

cosθ = 1/√(1+tan^2 θ)

But θ = tan^-1(u), so tanθ = u and we have

cosθ = 1/√(1+u^2)

I have always easier to carefully draw the triangle and label the sides. Then all the trig functions just drop right out.