If the period of the function $f(x)=\frac{\tan 5 x \cos 3 x}{\sin 6 x}$ is $\alpha$, then $(\alpha)$…

If the period of the function $f(x)=\frac{\tan 5 x \cos 3 x}{\sin 6 x}$ is $\alpha$, then $(\alpha)$ $f\left(\frac{\boldsymbol{\alpha}}{8}\right)=$
  1. $\frac{1}{2}$
  2. -1
  3. $\frac{1}{\sqrt{2}}$
  4. $-\frac{1}{\sqrt{2}}$

Solution

Period of $\tan 5 x, \cos 3 x, \sin 6 x$ is $\frac{\pi}{5}, \frac{2 \pi}{3}, \frac{\pi}{3}$ respectively.
Period of $f(x)=\frac{2 \pi}{1}=\alpha \Rightarrow \frac{\alpha}{8}=\frac{\pi}{4}$ $f\left(\frac{\alpha}{8}\right)=\frac{\tan \left(\frac{5 \pi}{4}\right) \cos \left(\frac{3 \pi}{4}\right)}{\sin \left(\frac{3 \pi}{2}\right)}=\frac{-1}{\sqrt{2}} \times(-1)=\frac{1}{\sqrt{2}}$

Asked in: AP EAMCET 2024 (21 May Shift 1)

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