Question

$\cos^{-1}\left(\frac{4}{5}\right)+\cos^{-1}\left(\frac{12}{13}\right)=$

• A. $\cos^{-1}\left(\frac{24}{25}\right)$
• B. $\cos^{-1}\left(\frac{33}{65}\right)$
• C. $\cos^{-1}\left(\frac{5}{13}\right)$
• D. $\cos^{-1}\left(\frac{3}{5}\right)$

Hint:

You can also solve this by converting the inverse cosines to inverse tangents using Pythagorean triplets: $\cos^{-1}(4/5) = \tan^{-1}(3/4)$ and $\cos^{-1}(12/13) = \tan^{-1}(5/12)$. Then apply the $\tan^{-1}A + \tan^{-1}B$ formula!

Answer 1

The Correct Option is B

Step 1: Name the two angles.
Let $A = \cos^{-1}\frac{4}{5}$ and $B = \cos^{-1}\frac{12}{13}$. So $\cos A = \frac{4}{5}$ and $\cos B = \frac{12}{13}$.

Step 2: Find the sines.
Using $\sin = \sqrt{1 - \cos^2}$: $\sin A = \frac{3}{5}$ and $\sin B = \frac{5}{13}$.

Step 3: Use the cosine addition formula.
\[ \cos(A + B) = \cos A\cos B - \sin A\sin B \]
Step 4: Put the values in.
\[ \cos(A+B) = \frac{4}{5}\cdot\frac{12}{13} - \frac{3}{5}\cdot\frac{5}{13} = \frac{48}{65} - \frac{15}{65} \]
Step 5: Subtract.
\[ \cos(A+B) = \frac{33}{65} \]
Step 6: Write the answer.
So $A + B = \cos^{-1}\frac{33}{65}$. \[ \boxed{\cos^{-1}\frac{33}{65} \text{ (Option 2)}} \]