algebra precalculus - Find $sin(x+y)$, given $tan x$ and $cos y$

Given that $\tan x= -2$ and $\cos y= 1/2$ where $x$ and $y$ are in the 4th and 1st quadrants respectively. Find, without evaluating angles $x$ and $y$,

a) $\sin (x+y)$

Here is what i have done so far..

For (X)

a² + b² = c²

1² + (-2)² = c²

1 +4 = c²

√5 = c²

For (Y)

a² + b² = c²

1² + b² = 2²

1 + b² = 2²

√b² = √3

√5 = c²

From here

sin (x+y)

sinx cosy + cosx sin y

= (-2/√5)(1/2) + (1/√5)(√3/2)

= (-2/2√5) + (1√3/2√5)

= -2+√3/√5 x √5/√5

= -2√5+√15/2√25

i got lost at this point

Answer

$$1+\tan^2 x=\sec^2 x$$ so
$\sec^2 x=5$ this means that $\cos x =\frac{1}{\sqrt{5}}$ since $x$ is fourth quadrent.

$$\sin x =\tan x \cos x = -\frac{2}{\sqrt{5}}$$

Also $\sin^2 y +\cos ^2 y=1$ gives $\sin y=\frac{\sqrt{3}}{2}$

So
$$\sin(x+y)=\cos(y)\sin(x)+\cos(x)\sin(y)=-\frac{2}{\sqrt{5}}\frac{1}{2}
+\frac{1}{\sqrt{5}}\frac{\sqrt{3}}{2}=\frac{\sqrt{3}-2}{2\sqrt{5}}$$