Question

If $\left(\sin x\right)\left(\cos y\right)$ $=\frac{1}{2}$ , then $\frac{d^{2}y}{d{x}^2}$ at $(\frac{\pi }{4},\frac{\pi }{4})$ is equal to

• A. $-4$
• B. $-2$
• C. $-6$
• D. $0$

Answer 1

The correct option is A $-4$

$\cos x\cos y-\sin x\sin y\frac{dy}{dx}=0$

$\frac{dy}{dx}=\cot x\cot y$

$\Rightarrow \frac{dy}{dx}{|}_{(\frac{\pi }{4},\frac{\pi }{4})}=1$

$\frac{d^{2}y}{d{x}^2}=-{\text{cosec}}^{2}x\cot y-\cot x{\text{cosec}}^{2}y\frac{dy}{dx}$

$\Rightarrow \frac{d^{2}y}{d{x}^2}{|}_{(\frac{\pi }{4},\frac{\pi }{4})}=-\left(2\right)\left(1\right)-\left(2\right)\left(1\right)\left(1\right)=-4$