Maximising the argument of $\sin and \cos$, given a linear relation between them

Question

Here is a problem I have been struggling with for a while,

If $$4\sin\theta \cos\phi+2\sin\theta+2\cos\phi+1=0$$

where $\theta,\phi\in[0,2\pi]$,

find the largest possible value of $(\theta+\phi)$

I have no idea how to do it, i tried substituting $r\sin\theta=a$ and $r\cos\phi=b$ but this did not worked out effectively. Any other approach for it?

Answer 1

Hint:

We have $$4\cos\phi\sin\theta+2\cos\phi+2\sin\theta+1=(2\cos\phi+1)(2\sin\theta+1)$$

Use all sin tan cos rule