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To solve \( \sin \left( \cos^{-1} \left( \frac{3}{4} \right) - \tan^{-1} \left( \frac{1}{4} \right) \right) \), let's go through each component step-by-step.

### Step 1: Set up angles
1. Let \( \theta = \cos^{-1} \left( \frac{3}{4} \right) \).
   - Then, \( \cos \theta = \frac{3}{4} \).
   - Using the Pythagorean identity, we can find \( \sin \theta \):
     \[
     \sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left( \frac{3}{4} \right)^2} = \sqrt{1 - \frac{9}{16}} = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4}.
     \]
   - Therefore, \( \sin(\theta) = \frac{\sqrt{7}}{4} \) and \( \cos(\theta) = \frac{3}{4} \).

2. Let \( \phi = \tan^{-1} \left( \frac{1}{4} \right) \).
   - Then, \( \tan \phi = \frac{1}{4} \).
   - Using the Pythagorean identity, we can find \( \sin \phi \) and \( \cos \phi \):
     \[
     \sin \phi = \frac{1}{\sqrt{1 + \left( \frac{1}{4} \right)^2}} = \frac{1}{\sqrt{1 + \frac{1}{16}}} = \frac{1}{\sqrt{\frac{17}{16}}} = \frac{4}{\sqrt{17}},
     \]
     and
     \[
     \cos \phi = \frac{1}{\sqrt{1 + \tan^2 \phi}} = \frac{4}{\sqrt{17}}.
     \]

### Step 2: Use the Sine Difference Identity
Now we need to find \( \sin(\theta - \phi) \) using the identity:
\[
\sin(\theta - \phi) = \sin \theta \cos \phi - \cos \theta \sin \phi.
\]

Substitute the values we found:
\[
\sin(\theta - \phi) = \left( \frac{\sqrt{7}}{4} \right) \left( \frac{4}{\sqrt{17}} \right) - \left( \frac{3}{4} \right) \left( \frac{1}{\sqrt{17}} \right).
\]

Simplify each term:
\[
= \frac{4 \sqrt{7}}{4 \sqrt{17}} - \frac{3}{4 \sqrt{17}} = \frac{4 \sqrt{7} - 3}{4 \sqrt{17}}.
\]

### Final Answer
The exact value of the expression is:
\[
\frac{4 \sqrt{7} - 3}{4 \sqrt{17}}.
\]

Would you like further details on any step? 

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Here are some related questions that can help expand your understanding:
1. How can we find exact values for inverse trigonometric functions without a calculator?
2. Why do we use the Pythagorean identity in this problem?
3. What are some applications of trigonometric identities in solving real-world problems?
4. How do we simplify expressions with square roots in trigonometry?
5. What are the differences between the sine and cosine of an angle, and how do they affect the results?

**Tip:** Remember that trigonometric identities are powerful tools for simplifying complex expressions, especially when involving inverse trigonometric functions.

Find the exact value of the expression \( \sin \left( \cos^{-1} \left( \frac{3}{4} \right) - \tan^{-1} \left( \frac{1}{4} \right) \right) \).

Learn how to find the exact value of \( \sin \left( \cos^{-1} \left( \frac{3}{4} \right) - \tan^{-1} \left( \frac{1}{4} \right) \) using trigonometric identities and inverse trigonometric functions. This solution covers the Pythagorean identity and the sine difference identity, ideal for students in Grades 11-12.

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