Mathematics - Trigonometric Equation Question with Solution | TestHub

The first equation can be written as

$2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)=2\sin \left(\frac{x+y}{2}\right)\cos \left(\frac{x+y}{2}\right)$

or $2\sin \left(\frac{x+y}{2}\right)\left[\cos \left(\frac{x-y}{2}\right)-\cos \left(\frac{x+y}{2}\right)\right]=0$

or $2\sin \left(\frac{x+y}{2}\right)2\sin \left(\frac{x}{2}\right)\sin \left(\frac{y}{2}\right)=0$.

Either $\frac{x+y}{2}=n\pi$

or $\frac{x}{2}=n\pi$ or $\frac{y}{2}=n\pi$.

Either $x+y=2n\pi$ or

$x=2n\pi$ or $y=2n\pi$.

Or $y=2n\pi$.

$|x|+|y|=1$.

$x\le 1$

and $y\le 1$.

Hence, $x+y=0$.

or x = 0 or y = 0 clearly y = 0 cuts the curve

$\left|x\right|+\left|y\right|=1$ at A, C, x = 0, cuts the curve

$\left|x\right|+\left|y\right|=1$ at B, D and x + y = 0 cuts the curve at E, F, hence 6 solutions are possible