(a) In a right angled triangle , the difference between two acute angles is `(pi)/(5)` in radians Express the angles in degrees.
(b) Evaluate : `6"cos"(pi)/(9)-8"cos"^(3)pi/(9)`

Answer

### Solution:

**(a)** In a right-angled triangle, the difference between two acute angles is given as \(\frac{\pi}{5}\) radians. We need to express these angles in degrees.

1. **Identify the angles**: Let the two acute angles be \(A\) and \(C\). We know that:
\[ A + C = 90^\circ \quad \text{(since the sum of angles in a triangle is 180° and one angle is 90°)} \]
\[ A - C = \frac{\pi}{5} \quad \text{(given)} \]

2. **Convert radians to degrees**: Convert \(\frac{\pi}{5}\) radians to degrees:
\[ \frac{\pi}{5} \text{ radians} = \frac{180^\circ}{5} = 36^\circ \]

3. **Set up the equations**: Now we have the following two equations:
\[ A + C = 90^\circ \quad \text{(1)} \]
\[ A - C = 36^\circ \quad \text{(2)} \]

4. **Add the equations**: Adding equations (1) and (2):
\[ (A + C) + (A - C) = 90^\circ + 36^\circ \]
\[ 2A = 126^\circ \]
\[ A = \frac{126^\circ}{2} = 63^\circ \]

5. **Substitute to find \(C\)**: Substitute \(A\) back into equation (1) to find \(C\):
\[ 63^\circ + C = 90^\circ \]
\[ C = 90^\circ - 63^\circ = 27^\circ \]

Thus, the angles are:
- \(A = 63^\circ\)
- \(C = 27^\circ\)

**(b)** Now we need to evaluate:
\[ 6 \cos\left(\frac{\pi}{9}\right) - 8 \cos^3\left(\frac{\pi}{9}\right) \]

1. **Factor the expression**: We can factor out \(-2\):
\[ 6 \cos\left(\frac{\pi}{9}\right) - 8 \cos^3\left(\frac{\pi}{9}\right) = -2(4 \cos^3\left(\frac{\pi}{9}\right) - 3 \cos\left(\frac{\pi}{9}\right)) \]

2. **Use the cosine triple angle formula**: Recall the formula:
\[ \cos(3\theta) = 4 \cos^3(\theta) - 3 \cos(\theta) \]
Here, let \(\theta = \frac{\pi}{9}\):
\[ \cos\left(3 \cdot \frac{\pi}{9}\right) = \cos\left(\frac{\pi}{3}\right) \]
Thus,
\[ 4 \cos^3\left(\frac{\pi}{9}\right) - 3 \cos\left(\frac{\pi}{9}\right) = \cos\left(\frac{\pi}{3}\right) \]

3. **Evaluate \(\cos\left(\frac{\pi}{3}\right)\)**: We know that:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

4. **Substitute back**: Now substitute this back into our expression:
\[ -2 \left(\frac{1}{2}\right) = -1 \]

Thus, the final answer is:
\[ 6 \cos\left(\frac{\pi}{9}\right) - 8 \cos^3\left(\frac{\pi}{9}\right) = -1 \]