Question

Show that f(x) = x/sin x is increasing on ]0, π/2[.

Answer 1

It is given that

f(x) = x/sin x

By differentiating w.r.t. x

f’(x) = (sin x. 1 – x. cos x)/(sin x)² = (sin x – x cos x)/(sin x)²

We know that

f’(x) > 0

By substituting the values

(sin x – x cos x)/(sin x)² > 0

So we get

(sin x – x cos x) > 0

Here tan x > x which is true on ] 0, π/2[.