Let  f(x)=cosx(sinx+sin2x+sin2θ) where θ is a given constant, then maximum value of f(x) is

f(x)=cosx(sinx+sin2+sin2θ)⇒(f(x)secx−sinx)2=sin2x+sin2θ⇒f2(x)sec2x−2f(x)tanx=sin2θ⇒f2(x)tan2x−2f(x)tanx+f2(x)−sin2θ=0∴4f2(x)≥4f2(x)(f2(x)−sin2θ)⇒f2(x)≤1+sin2θ⇒|f(x)≤1+sin2θ|