Question

If $\mathrm{y}={\sec }^{-1}[\mathrm{cosec}\mathrm{x}]+{\mathrm{cosec}}^{-1}[\sec \mathrm{x}]+{\sin }^{-1}[\cos \mathrm{x}]+{\cos }^{-1}[\sin \mathrm{x}],\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• A. $0$
• B. $2$
• C. $-2_{}$
• D. $-4$

Answer 1

The correct option is D

$-4$

Find the derivative:

Step1: Simplification of y

Here,

$\begin{array}{rcl}\mathrm{y}& =& {\sec }^{-1}[\mathrm{cosec}\mathrm{x}]+{\mathrm{cosec}}^{-1}[\sec \mathrm{x}]+{\sin }^{-1}[\cos \mathrm{x}]+{\cos }^{-1}[\sin \mathrm{x}]\\ & =& {\sec }^{-1}\left[\sec \right(90-\mathrm{x}\left)\right]+{\mathrm{cosec}}^{-1}\left[\mathrm{cosec}\right(90-\mathrm{x}\left)\right]+{\sin }^{-1}\left[\sin \right(90-\mathrm{x}\left)\right]+{\cos }^{-1}\left[\right(\cos (90-\mathrm{x})]\\ & =& (90-\mathrm{x})+(90-\mathrm{x})+(90-\mathrm{x})+(90-\mathrm{x})\\ & =& 4\times (90-\mathrm{x})\end{array}$

Step2: Find the $\frac{dy}{dx}$

$y=4(90-x)$

Differentiate with respect to $x$.

$$\begin{gathered} \frac{dy}{dx}=-4\frac{dx}{dx} \\ =-4 \end{gathered}$$

Hence, D is the correct option.