Question

Prove that cos(sinx) > sin(cos x), if x ∈ (0, π/2).

Answer 1

Let f(x) = x – sinx

⇒ f'(x) = 1 – cos x > 0, ∀ x ∈ (0, π/2) 

Thus, f(x) is strictly increasing in (0, π/2)

⇒ f(x) > f(0)

⇒ x – sinx > 0

⇒ x > sinx

⇒ cos x < cos(sinx) ...(i)

Also, for all x in (0, π/2), 0 < cos x < 1

⇒ cos x < 1 ⇒ cos x > sin (cos x) ...(ii)

From (i) and (ii), we get,

sin (cosx) < cosx < cos(sinx)