Question
For any real θ, find the maximum value of cos^2(cosθ) + sin^2(sinθ).
Answer
cos^2(cosθ) + sin^2(sinθ)
=cos^2(cosθ) + sin^2(sinθ)+ sin^2(cosθ)- sin^2(cosθ)
=1+sin^2(sinθ)- sin^2(cosθ)
The maximum value of cos^2(cosθ) + sin^2(sinθ) is 1+sin^2(sin 1), which occurs when cosθ=0, sinθ=1