<s>[INST] Maximum value of trigonometric equation $\cos^2(\cos\theta) + \sin^2(\sin\theta)$ For any real $\theta$ the maximum value of $$\cos^2(\cos\theta) + \sin^2(\sin\theta)$$
A. $1$
B. $1 + \sin^21$
C. $1 + \cos^21$
D. does not exist
I tried it by converting the whole expression into $\sin$ but getting nowhere with that. 
$$1-\sin^2(\cos\theta) + \sin^2(\sin\theta)$$
Now since 1 is constant therefore, $$[\sin^2(\cos\theta) + \sin^2(\sin\theta)]$$ should be minimum but I don't know how to minimize it. 
Also is there a way to think about it's solution graph. 
I have to solve this without using calculus.
Kindly help.
 [/INST] $$\max_{\theta\in\mathbb{R}}\{\cos^2(\cos\theta) + \sin^2(\sin\theta)\}\le\max_{-1\le x \le 1}\cos^2 x+\max_{-1\le y \le 1}\sin^2 y=1+\sin^21$$
$$\cos^2 \left( \cos \frac{\pi}{2} \right) + \sin^2 \left( \sin \frac{\pi}{2} \right) = 1 + \sin^2 1$$
So, the maximum is at most $1+\sin^21$, and this value is achieved. Hence the answer is b).
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