<s>[INST] If $|\sin^2 x+ 17 - x^2|=|16-x^2|+2 \sin ^2 x + \cos ^2x$ then $x$ lies in what interval If $|\sin^2 x+ 17 - x^2|=|16-x^2|+2 \sin ^2 x + \cos ^2x~$  then $x$ lies in what interval?
Hints Please? I dont know how to approach this problem. 
 [/INST] Observe that $|\sin^2x+1+16-x^2|=|16-x^2|+\sin^2x+1$ $(*)$
Set $y=\sin^2x+1$ and $z=16-x^2$. Then $(*)$ reduce to
$|z+y|=|z|+y$. Taking square of both sides we get $y(z-|z|)=0$
Hence, since $\sin^2x+1\neq0$ we have $16-x^2=|16-x^2|$ and this is true only for $ -4\leq x\leq 4$.
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