Question

\(\rm{tan} \dfrac{5 \pi}{12} - \rm{tan} \dfrac{\pi}{12} - \sqrt 3 \rm{tan} \dfrac{5 \pi}{12}. \rm{tan} \dfrac{ \pi}{12}\) is equal to
1. √3
2. -√3
3. \(\dfrac{1}{\sqrt 3}\)
4. 1

Answer 1

Correct Answer - Option 1 : √3

Formula used:

\(tan(A - B) = \frac{tanA - tanB}{1 + tanA.tanB}\)

Calculation:

\(tan(A - B) = \frac{tanA - tanB}{1 + tanA.tanB}\)

\(\Rightarrow tan(\frac{5\pi}{12} - \frac{\pi}{12}) = \frac{tan(\frac{5\pi}{12} - tan \frac{\pi}{12})}{1 + tan\frac{5\pi}{12}.tan \frac{\pi}{12}}\)

\(\Rightarrow tan(\frac{\pi}{3}) = \frac{tan(\frac{5\pi}{12} - tan \frac{\pi}{12})}{1 + tan\frac{5\pi}{12}.tan \frac{\pi}{12}}\)

\(\Rightarrow \sqrt3 = \frac{tan(\frac{5\pi}{12} - tan \frac{\pi}{12})}{1 + tan\frac{5\pi}{12}.tan \frac{\pi}{12}}\)

\(\Rightarrow \sqrt 3(1 + tan\frac{5\pi}{12}.tan \frac{\pi}{12}) = {tan(\frac{5\pi}{12} - tan \frac{\pi}{12})}{}\)

\(\Rightarrow {tan(\frac{5\pi}{12} - tan \frac{\pi}{12})} - \sqrt 3 .tan\frac{5\pi}{12}.tan \frac{\pi}{12}) = \sqrt 3\)