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Question: Solution of the equation \(\cos x \cos y \frac { d y } { d x } = - \sin x \sin y\)is...

Solution of the equation cos⁡xcos⁡ydydx=−sin⁡xsin⁡y\cos x \cos y \frac { d y } { d x } = - \sin x \sin yis

A

sin⁡y+cos⁡x=c\sin y + \cos x = c

B

sin⁡y−cos⁡x=c\sin y - \cos x = c

C

sin⁡y⋅cos⁡x=c\sin y \cdot \cos x = c

D

sin⁡y=ccos⁡x\sin y = c \cos x

Answer

sin⁡y=ccos⁡x\sin y = c \cos x

Explanation

Solution

cos⁡xcos⁡ydydx=−sin⁡xsin⁡y\cos x \cos y \frac { d y } { d x } = - \sin x \sin y

⇒ cos⁡ysin⁡ydy=−sin⁡xcos⁡xdx\frac { \cos y } { \sin y } d y = - \frac { \sin x } { \cos x } d x ⇒ cot⁡ydy=−tan⁡xdx\cot y d y = - \tan x d x

On integrating, we get

log⁡sin⁡y=log⁡cos⁡x+log⁡c\log \sin y = \log \cos x + \log c ⇒ sin⁡y=ccos⁡x\sin y = c \cos x .