Solving the system $\tan x + \tan y = 1$ and $\cos x \cdot \sin y = \frac{\sqrt{2}}{2}$ How can I solve this system of trigonometric equations:
$$\tan x + \tan y = 1$$
$$\cos x \cdot \sin y = \frac{\sqrt{2}}{2}$$
I tried to write tangent as $\sin/\cos$ and then multiply the first equation with the second one but it is not that brought me in a right way.
Do you have any idea how to solve this one?

$$\cos x \sin y=\frac{\sqrt2}{2}$$
$$\cos x \sin y=\frac{1}{\sqrt2}$$
$\frac{1}{\sqrt2}$ can be written as $1*\frac{1}{\sqrt2}$.
This gives rise to two cases-->

*

*$\cos x=1$ and $\sin y=\frac{1}{\sqrt2}$. So, $x=0$ and $y=\frac{\pi}{4}$.

*$\cos x=\frac{1}{\sqrt2}$ and $\sin y=1$. So, $x=\frac{\pi}{4}$ and $y=\frac{\pi}{2}$.

Solving Equation 1 for both cases-->

*

*$$\tan x+\tan y$$
$$\tan 0+\tan\frac{\pi}{4}$$
$$0+1=1$$
2.$$\tan x+\tan y$$
$$\tan\frac{\pi}{4}+\tan\frac{\pi}{2}$$
$$1+\infty=\infty$$
We see that only Case 1 satisfies the given equation. So, $x=0$ and $y=\frac{\pi}{4}$.
