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

Answer

`y-0=-frac{1}{2}(x-frac{pi}{2})`

and


`y-0=-frac{1}{2}(x+frac{3 pi}{2})`

Explanation

ATQ, the given equation is `x+2y=0`
$$
\begin{aligned}
&\text { Slope }=-\frac{1}{2} \\ \\
&=-\frac{1}{2}=-\sin (x+y)(1+y) \\ \\
&\Rightarrow-\frac{1}{2}=-\sin (x+y)\left(1-\frac{1}{2}\right) \\ \\
&\Rightarrow \sin (x+y)=1 \\ \\
&\Rightarrow \cos (x+y)=0 \Rightarrow y=0 \cos x=0 \\ \\
...