Prove that f(x) = tan-1 (sin x + cos x), x > 0 is always an increasing

1.	Prove that f(x) = tan-1 (sin x + cos x), x > 0 is always an increasing function in (0,π/4).
Answer» Given that, f(x) = tan^-1(sin x + cos x), x > 0 ∴ f(x) = 1/(1 + (sin x + cos x)²).(cos x - sin x) = cos x/(1 + (sin x + cos x)²).(1 - tan x) > 0, ∀ x ∈ (0,π/4) Hence, f(x) > 0 in (0,π/4) Therefore f(x) is an increasing function is (0,π/4)

Prove that f(x) = tan-1 (sin x + cos x), x > 0 is always an increasing function in (0,π/4).

Answer»

Given that, f(x) = tan^-1(sin x + cos x), x > 0

∴ f(x) = 1/(1 + (sin x + cos x)²).(cos x - sin x)

= cos x/(1 + (sin x + cos x)²).(1 - tan x) > 0, ∀ x ∈ (0,π/4)

Hence, f(x) > 0 in (0,π/4)

Therefore f(x) is an increasing function is (0,π/4)