Question

What is the value of $\cos \left\{\begin{array}{c}{\cos }^{-1}\frac{4}{5}+{\cos }^{-1}\frac{12}{13}\end{array}\right\}$ ?

• A. 63/65
• B. 33/65
• C. 22/65
• D. 11/65

Answer 1

The correct option is B 33/65

Now,

${\cos }^{-1}a+{\cos }^{-1}b={\cos }^{-1}(ab-\sqrt{(1-a^2)(1-b^2)})$

$\therefore {\cos }^{-1}\frac{4}{5}+{\cos }^{-1}\frac{12}{13}={\cos }^{-1}(\frac{48}{65}-\sqrt{(1-\frac{16}{25})(1-\frac{144}{169})})$

$={\cos }^{-1}\left(\frac{48-15}{65}\right)={\cos }^{-1}\left(\frac{33}{65}\right)$

$\cos \left({\cos }^{-1}\left(\frac{33}{65}\right)\right)=\frac{33}{65}$

Hence, B is correct.