Question

If α = ${\tan }^{-1}\left(\tan \frac{5\mathrm{\pi }}{4}\right)\mathrm{and}\mathrm{\beta }={\tan }^{-1}\left(-\tan \frac{2\mathrm{\pi }}{3}\right)$, then
(a) 4 α = 3 β
(b) 3 α = 4 β
(c) α − β = $\frac{7\mathrm{\pi }}{12}$
(d) none of these

Answer 1

(a) 4 α = 3 β

We know that ${\tan }^{-1}\left(\tan x\right)=x$.

$$\begin{gathered} \therefore \alpha ={\tan }^{-1}\left(\tan \frac{5\mathrm{\pi }}{4}\right) \\ ={\tan }^{-1}\left\{\tan \left(\mathrm{\pi }+\frac{\mathrm{\pi }}{4}\right)\right\} \\ ={\tan }^{-1}\left(\tan \frac{\mathrm{\pi }}{4}\right) \\ =\frac{\mathrm{\pi }}{4} \end{gathered}$$
and
$$\begin{gathered} \beta ={\tan }^{-1}\left\{-\tan \left(\frac{2\mathrm{\pi }}{3}\right)\right\} \\ ={\tan }^{-1}\left\{-\tan \left(\mathrm{\pi }-\frac{\mathrm{\pi }}{3}\right)\right\} \\ ={\tan }^{-1}\left\{\tan \left(\frac{\mathrm{\pi }}{3}\right)\right\} \\ =\frac{\mathrm{\pi }}{3} \end{gathered}$$

$$\begin{gathered} \therefore 4\alpha =\pi \\ 3\beta =\pi \end{gathered}$$
∴ $4\alpha =3\beta$