Question

The value of $\tan 75^{\circ }-\cot 75^{\circ }$ is

• A. $2\sqrt{3}$
• B. $2-\sqrt{3}$
• C. $2+\sqrt{3}$
• D. 0

Answer 1

Answer: (A)

$\tan 75^{\circ }-\cot 75^{\circ }$
$=\frac{\sin 75^{\circ }}{\cos 75^{\circ }}-\frac{\cos 75^{\circ }}{\sin 75^{\circ }}$
$=\frac{{\sin }^{2}75^{\circ }-{\cos }^{2}75^{\circ }}{\sin 75^{\circ }\cdot \cos 75^{\circ }}=\frac{-2\cos 150^{\circ }}{\sin 150^{\circ }}$
$=\frac{-2\cos \left(90^{\circ }+60^{\circ }\right)}{\sin \left(90+60^{\circ }\right)}$
$=\frac{2\cdot \frac{\sqrt{3}}{2}}{\frac{1}{2}}=2\sqrt{3}$