Q.Find the value of $\cot 75^\circ$.

Using the compound-angle formula $\tan(45^\circ+30^\circ)=2+\sqrt3$, so $\cot75^\circ=\tfrac{1}{2+\sqrt3}=2-\sqrt3$.

Apply the tangent compound-angle formula to $75^\circ = 45^\circ + 30^\circ$:

$\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}$

With $\tan45^\circ=1$ and $\tan30^\circ=\tfrac{1}{\sqrt3}$:

$\tan75^\circ = \frac{1 + \tfrac{1}{\sqrt3}}{1 - 1\cdot\tfrac{1}{\sqrt3}} = \frac{\sqrt3 + 1}{\sqrt3 - 1}$

Rationalise by multiplying by $\dfrac{\sqrt3+1}{\sqrt3+1}$:

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