<s>[INST] How to find cotangent? Need to find a $3\cot(x+y)$ if $\tan(x)$ and $\tan(y)$ are the solutions of $x^2-3\sqrt{5}\,x +2 = 0$.
I tried to solve this and got $3\sqrt{5}\cdot1/2$, but the answer is $-\sqrt{5}/5$
 [/INST] From quadratic equation given, 
$$\tan{x}+\tan{y}=3\sqrt5$$
$$\tan{x}\tan{y}=2$$
So by the Compound Angle Formula,
$$\tan{(x+y)}=\frac{\tan{x}+\tan{y}}{1-\tan{x}\tan{y}}=\frac{3\sqrt5}{-1}=-3\sqrt5$$
So we have
$$3\cot{(x+y)}=3\cdot{\frac1{\tan{(x+y)}}}=3\cdot{\frac1{-3\sqrt5}}=\frac{-\sqrt5}{5}$$
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