Question

Which of the following option(s) is/are correct for $0<x<\frac{\pi }{2}$

• A. $\cos \left(\sin x\right)>\cos x$
• B. $\cos \left(\sin x\right)>\sin \left(\cos x\right)$
• C. $\sin \left(\cos x\right)<\cos x$
• D. $\sin \left(\cos x\right)>\cos \left(\sin x\right)$

Answer 1

Answer: (A), (B), (C)

$\sin \left(\cos x\right)<\cos x$

We know $\sin x<x,\forall x\in (0,\frac{\pi }{2})$
$\Rightarrow \cos \left(\sin x\right)>\cos x\cdots \left(i\right)$
$\{\because \cos x\text{is decreasing in}(0,\frac{\pi }{2})\}$
Also $\sin \left(\cos x\right)<\cos x\cdots \left(ii\right)$
$\{\because \cos x\in (0,1\left)\right\}$
From $\left(i\right)$ and $\left(ii\right)$ we have
$\sin \left(\cos x\right)<\cos x<\cos \left(\sin x\right)$
$\Rightarrow \cos \left(\sin x\right)>\sin \left(\cos x\right)$