Question

The value of, $\tan {75}^0+\tan {15}^0=$

• A. $4$
• B. `$0$
• C. $1$
• D. $\infty$

Answer 1

The correct option is A $4$

$\tan {75}^{\circ }+\tan {15}^{\circ }$

$=\tan {75}^{\circ }+\tan ({90}^{\circ }-{75}^{\circ })$

$=\tan {75}^{\circ }+\cot {75}^{\circ }$

$=\tan {75}^{\circ }+\frac{1}{\tan {75}^{\circ }}$

$=\frac{{\tan }^2{75}^{\circ }+1}{\tan {75}^{\circ }}$

$=\frac{{\sec }^2{75}^{\circ }}{\tan {75}^{\circ }}$

$=\frac{\frac{1}{{\cos }^2{75}^{\circ }}}{\frac{\sin {75}^{\circ }}{\cos {75}^{\circ }}}$

$=\frac{1}{{\cos }^2{75}^{\circ }}\times \frac{\cos {75}^{\circ }}{\sin {75}^{\circ }}$

$=\frac{1}{\sin {75}^{\circ }\cos {75}^{\circ }}$

$=\frac{2}{2\sin {75}^{\circ }\cos {75}^{\circ }}$

$=\frac{2}{\sin {150}^{\circ }}$

$=\frac{2}{\sin (180-30)}$

$=\frac{2}{\sin {30}^{\circ }}$

$=\frac{2}{\frac{1}{2}}=4$