Find all the tangents to the curve `y=cos(x+y),-2pilt=xlt=2pi` that are parallel to the line `x+2y=0.`

Answer

`rightarrow quad x+2 y=frac{pi}{2}`
and `x+2 h=-frac{3 pi}{2}`

are the required equations of tangents.

Explanation

ATQ,

`y=\cos (x+y)`


`\Rightarrow(\frac{d y}{d x})=-\sin (x+y) \cdot(1+\frac{d y}{d x})`


Since, tangent is parallel to `x+2 y=0`,

then slope `\frac{d y}{d x}=-\frac{1}{2}`

From Eq. (i), `-\frac{1}{2}=-\sin (x+y) \cdot(1-\frac{1}{2})`

`\Rightarrow \sin (x+y)=1`, which shows `\cos (x+y)=0`

`\therefore \quad y=0`

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