`cos^(-1)((4)/(5))+cos^(-1)((12)/(13))`=

Options

A. `cos^(-1)((48)/(65))`
B. `cos^(-1)((33)/(65))`
C. `cos^(-1)((11)/(65))`
D. `pi-cos^(-1)((8)/(65))`

Answer

To solve the expression \( \cos^{-1}\left(\frac{4}{5}\right) + \cos^{-1}\left(\frac{12}{13}\right) \), we can use the formula for the sum of two inverse cosines:

\[ \cos^{-1}(x) + \cos^{-1}(y) = \cos^{-1}(xy - \sqrt{(1-x^2)(1-y^2)}) \]

### Step 1: Identify \( x \) and \( y \)
Let:
- \( x = \frac{4}{5} \)
- \( y = \frac{12}{13} \)

### Step 2: Calculate \( xy \)
Now, calculate \( xy \):
\[ xy = \frac{4}{5} \cdot \frac{12}{13} = \frac{48}{65} \]

### Step 3: Calculate \( 1 - x^2 \) and \( 1 - y^2 \)
Next, we need to find \( 1 - x^2 \) and \( 1 - y^2 \):
\[ 1 - x^2 = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \]
\[ 1 - y^2 = 1 - \left(\frac{12}{13}\right)^2 = 1 - \frac{144}{169} = \frac{25}{169} \]

### Step 4: Calculate \( \sqrt{(1-x^2)(1-y^2)} \)
Now, compute \( \sqrt{(1-x^2)(1-y^2)} \):
\[ \sqrt{(1-x^2)(1-y^2)} = \sqrt{\frac{9}{25} \cdot \frac{25}{169}} = \sqrt{\frac{225}{4225}} = \frac{15}{65} \]

### Step 5: Substitute into the formula
Now substitute \( xy \) and \( \sqrt{(1-x^2)(1-y^2)} \) back into the formula:
\[ \cos^{-1}\left(\frac{4}{5}\right) + \cos^{-1}\left(\frac{12}{13}\right) = \cos^{-1}\left(\frac{48}{65} - \frac{15}{65}\right) \]
\[ = \cos^{-1}\left(\frac{48 - 15}{65}\right) = \cos^{-1}\left(\frac{33}{65}\right) \]

### Final Answer
Thus, we have:
\[ \cos^{-1}\left(\frac{4}{5}\right) + \cos^{-1}\left(\frac{12}{13}\right) = \cos^{-1}\left(\frac{33}{65}\right) \]

### Conclusion
The final answer is \( \cos^{-1}\left(\frac{33}{65}\right) \).

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