If

Question:

If $\alpha=\tan ^{-1}\left(\tan \frac{5 \pi}{4}\right)$ and $\beta=\tan ^{-1}\left(-\tan \frac{2 \pi}{3}\right)$, then

(a) $4 \alpha=3 \beta$

(b) $3 a=4 \beta$

(c) $\alpha-\beta=\frac{7 \pi}{12}$

(d) none of these

Solution:

(a) $4 \alpha=3 \beta$

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

$\therefore \alpha=\tan ^{-1}\left(\tan \frac{5 \pi}{4}\right)$

$=\tan ^{-1}\left\{\tan \left(\pi+\frac{\pi}{4}\right)\right\}$

$=\tan ^{-1}\left(\tan \frac{\pi}{4}\right)$

$=\frac{\pi}{4}$

and

$\beta=\tan ^{-1}\left\{-\tan \left(\frac{2 \pi}{3}\right)\right\}$

$=\tan ^{-1}\left\{-\tan \left(\pi-\frac{\pi}{3}\right)\right\}$

$=\tan ^{-1}\left\{\tan \left(\frac{\pi}{3}\right)\right\}$

$=\frac{\pi}{3}$

$\therefore 4 \alpha=\pi$

$3 \beta=\pi$

$\therefore 4 \alpha=3 \beta$

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