Question

$f\left(x\right)=\frac{\sin x}{\sqrt{1+{\tan }^{2}x}}+\frac{\cos x}{\sqrt{1+{\cot }^{2}x}}$ is constant in which of following interval

• A. $(0,\frac{\pi }{2})$
• B. $\left(\frac{\pi }{2},\pi \right)$
• C. $(\pi ,\frac{\pi }{2})$
• D. $\left(\frac{3\pi }{2},2\pi \right)$

Answer 1

Answer: (B), (D)

$\left(\frac{\pi }{2},\pi \right)$
$\left(\frac{3\pi }{2},2\pi \right)$

$f\left(x\right)=\frac{\sin \left(x\right)}{|\sec \left(x\right)|}+\frac{\cos \left(x\right)}{|cosec\left(x\right)|}$
$=\sin \left(x\right)|\cos \left(x\right)|+\cos \left(x\right)|\sin \left(x\right)|$.
Hence
$f\left(x\right)=0$, constant, when $\sin \left(x\right)>0$ and $\cos \left(x\right)<0$ or $\cos \left(x\right)>0$ and $\sin \left(x\right)<0$.
This happens in II quadrant and IV quadrant.