What is the value of cot (90 + 75°)?
-2 - √3
Using \(\cot(90^\circ+\theta)=-\tan\theta\):
\[\cot(90^\circ+75^\circ)=-\tan 75^\circ\]
Compute \(\tan 75^\circ\): \(\tan(45^\circ+30^\circ)=\dfrac{1+1/\sqrt 3}{1-1/\sqrt 3}=\dfrac{\sqrt 3+1}{\sqrt 3-1}=2+\sqrt 3\).
Therefore \(\cot(165^\circ)=-(2+\sqrt 3)=\mathbf{-2-\sqrt 3}\).
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