Evaluate : `cos(sin^(-1)(1/4)+sec^(-1)(4/3))`

Explanation

To evaluate the expression \( \cos(\sin^{-1}(1/4) + \sec^{-1}(4/3)) \), we can use the cosine addition formula and properties of inverse trigonometric functions. Here’s a step-by-step solution:

### Step 1: Identify the angles
Let:
- \( A = \sin^{-1}(1/4) \)
- \( B = \sec^{-1}(4/3) \)

### Step 2: Use the cosine addition formula
We can express \( \cos(A + B) \) using the cosine addition formula:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]

### Step 3: Calculate \( \cos A \) and \( \sin A \)
Since \( A = \sin^{-1}(1/4) \):
- By definition, \( \sin A = 1/4 \).
- To find \( \cos A \), we use the identity \( \cos A = \sqrt{1 - \sin^2 A} \):
\[ \cos A = \sqrt{1 - (1/4)^2} = \sqrt{1 - 1/16} = \sqrt{15/16} = \frac{\sqrt{15}}{4} \]

### Step 4: Calculate \( \cos B \) and \( \sin B \)
Since \( B = \sec^{-1}(4/3) \):
- By definition, \( \sec B = 4/3 \) implies \( \cos B = 3/4 \).
- To find \( \sin B \), we use the identity \( \sin B = \sqrt{1 - \cos^2 B} \):
\[ \sin B = \sqrt{1 - (3/4)^2} = \sqrt{1 - 9/16} = \sqrt{7/16} = \frac{\sqrt{7}}{4} \]

### Step 5: Substitute values into the cosine addition formula
Now we substitute \( \cos A \), \( \sin A \), \( \cos B \), and \( \sin B \) into the formula:
\[ \cos(A + B) = \left(\frac{\sqrt{15}}{4}\right) \left(\frac{3}{4}\right) - \left(\frac{1}{4}\right) \left(\frac{\sqrt{7}}{4}\right) \]
\[ = \frac{3\sqrt{15}}{16} - \frac{\sqrt{7}}{16} \]
\[ = \frac{3\sqrt{15} - \sqrt{7}}{16} \]

### Final Answer
Thus, the value of \( \cos(\sin^{-1}(1/4) + \sec^{-1}(4/3)) \) is:
\[ \frac{3\sqrt{15} - \sqrt{7}}{16} \]