• # question_answer The solution of the differential equation $\tan y\frac{dy}{dx}=\sin (x+y)+sin(x-y)$ is A) $\sec y-2\cos x=c$ B) $\sec y+2\cos x=c$ C) $\cos y-2\sin x=c$ D) $\sec y+2\sin x=c$

Solution :

[b]: Given, $\tan y\frac{dy}{dx}=\sin (x+y)+sin(x-y)$ $\tan y\frac{dy}{dx}=\sin (x+y)+\sin (x-y)$$=2\sin x\cos y$ $\Rightarrow$$\tan y\sec ydy=2\sin xdx$ $\Rightarrow$$\int_{{}}^{{}}{\tan y\sec ydy=2}\int_{{}}^{{}}{\sin xdx}$ $\Rightarrow$$\sec y=-2\cos x+c\Rightarrow \sec y+2\cos x=c$

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