Question

The value of $tan75^{\circ} - cot75^{\circ}$ is equal to

• A. $2\sqrt{3}$
• B. $2+\sqrt{3}$
• C. $2-\sqrt{3}$
• D. $1$

Answer 1

Answer: (A)

$tan75^{\circ}-cot75^{\circ}=tan\left(45^{\circ}+30^{\circ}\right)-cot\left(45^{\circ}+30^{\circ}\right)$

$=\frac{tan\,45^{\circ}+tan\,30^{\circ}}{1-tan\,45^{\circ}\,tan\,30^{\circ}}-\frac{cot\,45^{\circ}\,cot\,30^{\circ}-1}{cot\,45^{\circ}+cot\,30^{\circ}}$

$=\frac{1+\frac{1}{\sqrt{3}}}{1-\frac{1}{\sqrt{3}}}-\frac{\sqrt{3}-1}{1+\sqrt{3}}$

$=\frac{\left(\sqrt{3}+1\right)}{\left(\sqrt{3}-1\right)}-\frac{\sqrt{3}-1}{\sqrt{3}+1}$

$=\frac{\left(3+1+2\sqrt{3}\right)}{3-1}-\frac{\left(3+1-2\sqrt{3}\right)}{3-1}$

$=\frac{4\sqrt{3}}{2}=2\sqrt{3}$