Q141 / 200 · TS EAMCET 2002 · Mathematics
If $f(x)=\sin ^2\left(\frac{\pi}{8}+\frac{x}{2}\right)-\sin ^2\left(\frac{\pi}{8}-\frac{x}{2}\right)$, then the period of $f$ is
- A$\frac{\pi}{3}$
- B$\frac{\pi}{2}$
- C$\pi$
- Correct answerD$2 \pi$
Solution
We have,
$$
\begin{aligned}
& f(x)=\sin ^2\left(\frac{\pi}{8}+\frac{x}{2}\right)-\sin ^2\left(\frac{\pi}{8}-\frac{x}{2}\right) \\
& f(x)=\sin \left(\frac{\pi}{8}+\frac{x}{2}+\frac{\pi}{8}-\frac{x}{2}\right) \sin \left(\frac{\pi}{8}+\frac{x}{2}-\frac{\pi}{8}+\frac{x}{2}\right)
\end{aligned}
$$
$$
=\sin \frac{\pi}{4} \sin x=\frac{1}{\sqrt{2}} \sin x
$$
$\therefore$ Period of $f(x)$ is $2 \pi$
[ $\because$ Period of $\sin x$ is $2 \pi$ ]
Hence, the correct answer is option: 'D'.