Question

The value of $co{s}^{-1}\left(\frac{3}{5}\right)+co{s}^{-1}\left(\frac{12}{13}\right)=$

• A. $co{s}^{-1}\left(\frac{16}{65}\right)$
• B. $co{s}^{-1}\left(\frac{33}{65}\right)$
• C. $co{s}^{-1}\left(\frac{77}{85}\right)$
• D. $co{s}^{-1}\left(\frac{12}{13}\right)$

Answer 1

The correct option is A $co{s}^{-1}\left(\frac{16}{65}\right)$

We have,

${\cos }^{-1}\left(\frac{3}{5}\right)+{\cos }^{-1}\left(\frac{12}{13}\right)$

We know that

${\cos }^{-1}x+{\cos }^{-1}y={\cos }^{-1}\{xy-\sqrt{1-x^2}\sqrt{1-y^2}\}$

Therefore,

$={\cos }^{-1}\{\frac{3}{5}\times \frac{12}{13}-\sqrt{1-{\left(\frac{3}{5}\right)}^2}\sqrt{1-{\left(\frac{12}{13}\right)}^2}\}$

$={\cos }^{-1}\{\frac{36}{65}-\sqrt{1-\frac{9}{25}}\sqrt{1-\frac{144}{169}}\}$

$={\cos }^{-1}\{\frac{36}{65}-\sqrt{\frac{16}{25}}\sqrt{\frac{25}{169}}\}$

$={\cos }^{-1}\{\frac{36}{65}-\frac{4}{5}\times \frac{5}{13}\}$

$={\cos }^{-1}\{\frac{36}{65}-\frac{4}{13}\}$

$={\cos }^{-1}\left\{\frac{36-20}{65}\right\}$

$={\cos }^{-1}\left\{\frac{16}{65}\right\}$

Hence, this is the answer.