Question

If α = \[\tan^{- 1} \left( \tan\frac{5\pi}{4} \right) \text{ and }\beta = \tan^{- 1} \left( - \tan\frac{2\pi}{3} \right)\] , then

 

Options

• 4 α = 3 β

• 3 α = 4 β

• α − β = `(7pi)/12`

• none of these

Answer 1

(a) 4 α = 3 β
We know that 

\[\tan^{- 1} \left( \tan{x} \right) = x\]
\[\therefore \alpha = \tan^{- 1} \left( \tan\frac{5\pi}{4} \right)\]
\[ = \tan^{- 1} \left\{ \tan\left( \pi + \frac{\pi}{4} \right) \right\}\]
\[ = \tan^{- 1} \left( \tan\frac{\pi}{4} \right)\]
\[ = \frac{\pi}{4}\]
and
\[\beta = \tan^{- 1} \left\{ - \tan\left( \frac{2\pi}{3} \right) \right\}\]
\[ = \tan^{- 1} \left\{ - \tan\left( \pi - \frac{\pi}{3} \right) \right\}\]
\[ = \tan^{- 1} \left\{ \tan\left( \frac{\pi}{3} \right) \right\}\]
\[ = \frac{\pi}{3}\]
\[\therefore 4\alpha = \pi\]
\[3\beta = \pi\]
∴ \[4\alpha = 3\beta\]