<s>[INST] If $A+B+C=180^\circ, \cot A=2$ and $\cot B= \frac34$, what is $\sin C$? 
If $A+B+C=180^\circ, \cot A=2$ and $\cot B= \frac34$, what is $\sin C$?

$$\cot A = 2\\
\frac{\cos A }{ \sin A }= 2\\
\cot B = \frac34\\
\frac{\cos B }{ \sin B} = \frac34$$
What should I do with $A+B+C=180^\circ$?
 [/INST] You have 
$$ \sin C = \sin \big(180^\circ -(A+B)\big) = \sin (A+B) $$
Then, using angle-sum formulas to obtain
$$ \cot (A+B) = \frac{\cot A \cot B - 1}{\cot A + \cot B} = \frac{2}{11} $$
And 
$$ \csc^2 (A+B) = \cot^2 (A + B) + 1 = \frac{125}{121} $$
You know what to do next.
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