Question

Find the equations of tangents to the curve $y=\cos \left(x+y\right),x\in \left[-2\pi ,2\pi \right]$ which is parallel to line $x+2y=0$

Answer 1

$y=\cos (x+y)\frac{dy}{dx}=-\sin (x+y)(1+\frac{dy}{dx})\Rightarrow \frac{dy}{dx}=\frac{-\sin (x+y)}{1+\sin (x+y)}$

$\therefore \left(\frac{dy}{dx}\right)=\frac{-1}{2}$ (for tangent parallel to $x+2y=0$)

$\therefore \frac{-\sin (x+y)}{1+\sin (x+y)}=\frac{-1}{2}\Rightarrow (x,y)=(\frac{\pi }{2},0)$

$\therefore$ Equation of tangent $\Rightarrow (y-0)=\frac{-1}{2}(x-\frac{\pi }{2})$

$\Rightarrow 2y+x=\frac{\pi }{2}$