Question

exact value of tan(pie/12) + cot(pie/12)l​

Answer 1

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

We are aware of the Trigonometric formula that

$\displaystyle \: sin \: 2\theta \: = \frac{2tan \theta}{1 + {tan}^{2}\theta}$

TO DETERMINE

$\displaystyle \: tan \: \frac{\pi}{12} + cot \: \frac{\pi}{12}$

CALCULATION

$\displaystyle \: tan \: \frac{\pi}{12} + cot \: \frac{\pi}{12}$

$= \displaystyle \: tan \: \frac{\pi}{12} + \frac{1}{ tan\: \frac{\pi}{12}}$

$= \displaystyle \: \frac{ {tan}^{2} \: \frac{\pi}{12} + 1}{ tan\: \frac{\pi}{12}}$

$= 2 \times \displaystyle \: \frac{ {tan}^{2} \: \frac{\pi}{12} + 1}{ 2 \: tan\: \frac{\pi}{12}}$

$= 2 \times \displaystyle \: \frac{1}{ \frac{ 2 \: tan\: \frac{\pi}{12}}{1 + {tan}^{2} \: \frac{\pi}{12} }}$

$\displaystyle \: \: = 2 \times \frac{1}{sin \: \frac{2\pi}{12}}$

$\displaystyle \: \: = 2 \times \frac{1}{sin \: \frac{\pi}{6}}$

$= \displaystyle \: \: 2 \times \frac{1}{ \frac{1}{2} }$

$= 2 \times 2$

$= 4$