The numerical value of `tan (cos ^(-1) "" (4)/(5) + tan ^(-1)"" (2)/(3))` is

Options

A. `(6)/(17)`
B. `(17)/(6)`
C. `(15)/(6)`
D. `(6)/(15)`

Answer

To solve the problem \( \tan \left( \cos^{-1} \left( \frac{4}{5} \right) + \tan^{-1} \left( \frac{2}{3} \right) \right) \), we will follow these steps:

### Step 1: Define the angles
Let:
- \( \theta = \cos^{-1} \left( \frac{4}{5} \right) \)
- \( \phi = \tan^{-1} \left( \frac{2}{3} \right) \)

### Step 2: Find \( \tan(\theta) \)
Using the definition of cosine:
\[ \cos(\theta) = \frac{4}{5} \]
We can find \( \sin(\theta) \) using the Pythagorean identity:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
\[ \sin^2(\theta) + \left( \frac{4}{5} \right)^2 = 1 \]
\[ \sin^2(\theta) + \frac{16}{25} = 1 \]
\[ \sin^2(\theta) = 1 - \frac{16}{25} = \frac{9}{25} \]
\[ \sin(\theta) = \frac{3}{5} \]

Now, we can find \( \tan(\theta) \):
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \]

### Step 3: Find \( \tan(\phi) \)
From the definition of tangent:
\[ \phi = \tan^{-1} \left( \frac{2}{3} \right) \]
Thus, \( \tan(\phi) = \frac{2}{3} \).

### Step 4: Use the tangent addition formula
Now we need to find \( \tan(\theta + \phi) \):
\[ \tan(\theta + \phi) = \frac{\tan(\theta) + \tan(\phi)}{1 - \tan(\theta) \tan(\phi)} \]
Substituting the values we found:
\[ \tan(\theta + \phi) = \frac{\frac{3}{4} + \frac{2}{3}}{1 - \left( \frac{3}{4} \cdot \frac{2}{3} \right)} \]

### Step 5: Calculate the numerator
Finding a common denominator for the numerator:
\[ \frac{3}{4} + \frac{2}{3} = \frac{9}{12} + \frac{8}{12} = \frac{17}{12} \]

### Step 6: Calculate the denominator
Calculating the denominator:
\[ 1 - \left( \frac{3}{4} \cdot \frac{2}{3} \right) = 1 - \frac{6}{12} = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 7: Combine the results
Now substituting back into the tangent addition formula:
\[ \tan(\theta + \phi) = \frac{\frac{17}{12}}{\frac{1}{2}} = \frac{17}{12} \cdot 2 = \frac{34}{12} = \frac{17}{6} \]

### Final Result
Thus, the numerical value of \( \tan \left( \cos^{-1} \left( \frac{4}{5} \right) + \tan^{-1} \left( \frac{2}{3} \right) \right) \) is:
\[ \frac{17}{6} \]