Evaluate:

Question:

Evaluate:

(i) $\cot \left(\sin ^{-1} \frac{3}{4}+\sec ^{-1} \frac{4}{3}\right)$

(ii) $\sin \left(\tan ^{-1} x+\tan ^{-1} \frac{1}{x}\right)$ for $x<0$

(iii) $\sin \left(\tan ^{-1} x+\tan ^{-1} \frac{1}{x}\right)$ for $x>0$

(iv) $\cot \left(\tan ^{-1} a+\cot ^{-1} a\right)$

(v) $\cos \left(\sec ^{-1} x+\operatorname{cosec}^{-1} x\right),|x| \geq 1$

Solution:

(i)

$\cot \left(\sin ^{-1} \frac{3}{4}+\sec ^{-1} \frac{4}{3}\right)$

$=\cot \left(\sin ^{-1} \frac{3}{4}+\cos ^{-1} \frac{3}{4}\right) \quad\left[\because \sec ^{-1} x=\cos ^{-1} \frac{1}{x}\right]$

$=\cot \left(\frac{\pi}{2}\right) \quad\left[\because \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}\right]$

$=0$

(ii)

$\sin \left(\tan ^{-1} x+\tan ^{-1} \frac{1}{x}\right)=\sin \left[\tan ^{-1}(-x)+\tan ^{-1}\left(-\frac{1}{x}\right)\right] \quad[\because x<0]$

$=\sin \left[-\tan ^{-1}(x)-\tan ^{-1}\left(\frac{1}{x}\right)\right]$

$=\sin \left\{-\left[\tan ^{-1}(x)+\tan ^{-1}\left(\frac{1}{x}\right)\right]\right\}$

$=\sin \left[-\left(\tan ^{-1} x+\cot ^{-1} x\right)\right] \quad\left[\because \tan ^{-1} \frac{1}{x}=\cot ^{-1} x\right]$

$=-\sin \left(\tan ^{-1} x+\cot ^{-1} x\right)$

$=-\sin \left(\frac{\pi}{2}\right)$

$=-1$

(iii)

$\sin \left(\tan ^{-1} x+\tan ^{-1} \frac{1}{x}\right)$

$=\sin \left(\tan ^{-1} x+\cot ^{-1} x\right) \quad\left[\because \tan ^{-1} x=\cot ^{-1} \frac{1}{x}\right]$

$=\sin \left(\frac{\pi}{2}\right) \quad\left[\because \tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}\right]$

$=1$

(iv)

$\cot \left(\tan ^{-1} a+\cot ^{-1} a\right)$

$=\cot \left(\frac{\pi}{2}\right) \quad\left[\because \tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}\right]$

$=0$

(v)

$\cos \left(\sec ^{-1} x+\operatorname{cosec}^{-1} x\right)$

$=\cos \left(\frac{\pi}{2}\right) \quad\left[\because \sec ^{-1} x+\operatorname{cosec}^{-1} x=\frac{\pi}{2}\right]$

$=0$

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