Evaluate the following expression:
`tan [cos^(-1)(3/4)+sin^(-1)(3/4)-sec^(-1)3]`

Explanation

To evaluate the expression \( \tan \left[ \cos^{-1} \left( \frac{3}{4} \right) + \sin^{-1} \left( \frac{3}{4} \right) - \sec^{-1}(3) \right] \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ \tan \left[ \cos^{-1} \left( \frac{3}{4} \right) + \sin^{-1} \left( \frac{3}{4} \right) - \sec^{-1}(3) \right] \]

### Step 2: Use the Identity
We know that:
\[ \sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2} \]
Thus, we can rewrite:
\[ \cos^{-1} \left( \frac{3}{4} \right) + \sin^{-1} \left( \frac{3}{4} \right) = \frac{\pi}{2} \]

### Step 3: Substitute into the Expression
Substituting this into our expression gives:
\[ \tan \left[ \frac{\pi}{2} - \sec^{-1}(3) \right] \]

### Step 4: Use the Co-Function Identity
Using the identity \( \tan \left( \frac{\pi}{2} - x \right) = \cot(x) \), we have:
\[ \tan \left[ \frac{\pi}{2} - \sec^{-1}(3) \right] = \cot(\sec^{-1}(3)) \]

### Step 5: Find the Value of \( \cot(\sec^{-1}(3)) \)
Let \( \theta = \sec^{-1}(3) \). Then, by definition of secant:
\[ \sec(\theta) = 3 \implies \cos(\theta) = \frac{1}{3} \]
Using the Pythagorean identity:
\[ \sin^2(\theta) + \cos^2(\theta) = 1 \implies \sin^2(\theta) + \left( \frac{1}{3} \right)^2 = 1 \]
\[ \sin^2(\theta) + \frac{1}{9} = 1 \implies \sin^2(\theta) = 1 - \frac{1}{9} = \frac{8}{9} \]
\[ \sin(\theta) = \frac{2\sqrt{2}}{3} \]

### Step 6: Calculate \( \cot(\theta) \)
Now, we can find \( \cot(\theta) \):
\[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{\frac{1}{3}}{\frac{2\sqrt{2}}{3}} = \frac{1}{2\sqrt{2}} \]

### Step 7: Final Result
Thus, we have:
\[ \tan \left[ \cos^{-1} \left( \frac{3}{4} \right) + \sin^{-1} \left( \frac{3}{4} \right) - \sec^{-1}(3) \right] = \frac{1}{2\sqrt{2}} \]

### Final Answer
\[ \boxed{\frac{1}{2\sqrt{2}}} \]