The correct option is C g(θ) is increasing in x∈(π4,π2)
To find; Monotonicity of g(θ) in (0,π4) and (π4,π2)
g(θ)=f(sinθ)+f(cosθ)g′(θ)=f′(sinθ)×cosθ−f′(cosθ)×sinθ.
f′(sinθ)<0 , f′′(sinθ)>0 ∀ θ∈(0,π2)
Let x=sinθ, f′(x)<0 ∀ x∈(0,1)
∵θ∈(0,π2)⇒cosθ ∈(0,1)
f is decreasing in (0,1)
f′′(x)>0 ∀ x ∈ (0,1)
⇒f′ is increasing in (0,1)
g′(θ)=f′(sinθ)cosθ−f′(cosθ)sinθ
f′(sinθ)<0 , f′(cosθ)<0
In (0,π4)⇒cosθ>sinθ
f′(cosθ)>f′(sinθ) and f′<0
⇒g′(θ)<0
∴g(θ) is decreasing in x∈(0,π4)
In (π4,π2)⇒sinθ>cosθ
f′(sinθ)<f′(cosθ) and f′<0
⇒g′(θ)>0
g(θ) is increasing in x∈(π4,π2)