Question

$4\sin x\cdot \cos y+2\cos y+1=0$ where $x,y\in [0,2\pi ]$ find the largest possible value of the sum (x+y).

Answer 1

$4\sin x\cdot \cos y+2\cos y+1=0$ $x,y\in [0,2\pi ]$

$\Rightarrow 16{\sin }^{2}x{\cos }^{2}y=4{\cos }^{2}y+4\cos y+1$

$\Rightarrow 16{\cos }^{2}y-16{\cos }^{2}x{\cos }^{2}y=4{\cos }^{2}y+4\cos y+1$

$\Rightarrow 12{\cos }^{2}y-4\cos y-1=16{\cos }^{2}x{\cos }^{2}y$

$\Rightarrow -12-4\sec y-{\sec }^{2}y=16{\cos }^{2}x$

$\Rightarrow -{\sec }^{2}y-4\sec y+12=16{\cos }^{2}x$

Maximum possible values of $x$ and $y$ that satisfies equation are at $x=2\pi$ and $y=\frac{4\pi }{3}$

Maximum value of $x+y=\frac{10\pi }{3}$