Explanation:
(D) : Given,
\(\int_{-\pi}^{\pi} \sin x[f(\cos x)] d x\)
\(\because \quad \int_{-a}^{a} g(x) d x=0\), for odd function.
\(g(x)=\sin x[f(\cos x)]\)
\(g(-x)=\sin (-x)[f(\cos (-x))]\)
\(=-\sin x[f(\cos x)]\)
\(=-\mathrm{g}(\mathrm{x})\)
Hence, it is odd function
Then, \(\quad \int_{-\pi}^{\pi} g(x) d x=0\)
So, \(\quad \int_{-\pi}^{\pi} \sin x[f(\cos )]=0\)
Hence, option (d) is correct.