Prove that y = (4 sin θ)/(2 + cos θ) - θ is an increasing function if θ ∈ [0, π/2]

Question

Prove that y = \(\frac{4\,sin\,\theta}{2 + cos\,\theta} - \theta\) is an increasing function if θ ∈ [0, π/2]

(4 sin θ)/(2 + cos θ)

Answer 1

y = (4 sin θ)/(2 + cos θ) - θ

\(=\frac{(8\, cos\, \theta + 4)-(2 + cos\,\theta)}{(2 + cos\,\theta)^2}\)

Hence, y is an increasing function if θ ∈ [0, π/2]