Question

If ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{2}$ then $\frac{dy}{dx}$ is equal to

• A. $\frac{x}{y}$
• B. $-\frac{x}{y}$
• C. $\frac{y}{x}$
• D. $-\frac{y}{x}$

Answer 1

The correct option is B

$-\frac{x}{y}$

Explanation for the correct option.

Step 1: Simplify the equation

Given that, ${\sin }^{-1}x+{\sin }^{-1}y=\frac{\pi }{2}$

$$\begin{gathered} {\sin }^{-1}x=\frac{\pi }{2}-{\sin }^{-1}y \\ {\sin }^{-1}x={\cos }^{-1}y \\ {\cos }^{-1}\left(\sqrt{1-x^2}\right)={\cos }^{-1}y \end{gathered}$$

Equating angles:

$y=\sqrt{1-x^2}......\left(1\right)$

Step 2: Differentiate the above eqution

Now differentiate $y$ with respect to $x$.

$\begin{array}{rcl}\therefore \frac{dy}{dx}& =& \frac{-2x}{2\sqrt{1-x^2}}\\ & =& -\frac{x}{y}from[\mathrm{Equation}(1\left)\right]\end{array}$

Hence, option B is correct.