Question

The value of $\tan [\cos^{-1}\frac{3}{5} + \tan^{-1} \frac{2}{3}] $ is

• A. 18
• B. 2
• C. 9
• D. None of these

Answer 1

Answer: (A)

$\tan \left[\cos ^{-1} \frac{3}{5}+\tan ^{-1} \frac{2}{5}\right]$
$=\tan \left[\tan ^{-1} \frac{\sqrt{1-\left(\frac{3}{5}\right)^{2}}}{\frac{3}{5}}+\tan ^{-1} \frac{2}{3}\right]$
$\left(\because \cos ^{-1} x=\tan ^{-1} \frac{\sqrt{1-x^{2}}}{x}\right)$
$=\tan \left[\tan ^{-1} \frac{\sqrt{25-9}}{5} \times \frac{5}{3}+\tan ^{-1} \frac{2}{3}\right]$
$=\tan \left[\tan ^{-1} \frac{4}{3}+\tan ^{-1} \frac{2}{3}\right]$
$\left.=\tan \left[\tan ^{-1}\left\{\frac{\left(\frac{4}{3}+\frac{2}{3}\right)}{1-\left(\frac{4}{3}\right)\left(\frac{2}{3}\right)}\right\}\right]\right]^{-1}$
$=\tan \left[\tan ^{-1}\left\{\frac{6}{3} \times \frac{9}{1}\right\}\right]$
$=\tan \left[\tan ^{-1}(18)\right]$
$=18$