If \(x \neq 0\), then \(\frac{\sin (\pi+x) \cos \left(\frac{\pi}{2}+x\right) \tan \left(\frac{3…

If \(x \neq 0\), then \(\frac{\sin (\pi+x) \cos \left(\frac{\pi}{2}+x\right) \tan \left(\frac{3 \pi}{2}-x\right) \cot (2 \pi-x)}{\sin (2 \pi-x) \cos (2 \pi+x) \operatorname{cosec}(-x) \sin \left(\frac{3 \pi}{2}+x\right)}=\)
  1. 0
  2. -1
  3. 1
  4. 2

Solution

\(\begin{aligned} & \frac{\sin (\pi+x) \cos \left(\frac{\pi}{2}+x\right) \tan \left(\frac{3 \pi}{2}-x\right) \cot (2 \pi-x)}{\sin (2 \pi-x) \cos (2 \pi+x) \operatorname{cosec}(-x) \sin \left(\frac{3 \pi}{2}+x\right)} \\ & (x \neq 0) \\ & =\frac{(-\sin x)(-\sin x)(\cot x)(-\cot x)}{(-\sin x)(\cos x)(-\operatorname{cosec} x)(-\cos x)}=1 \end{aligned}\) Hence, option (c) is correct.

Asked in: AP EAMCET 2020 (21 Sep Shift 1)

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