If cos 3x= 0 and x is acute find the value of :
` cot^(2) x- cosec ^(2)x`

Answer

To solve the problem, we need to follow these steps:

### Step 1: Solve for \( x \) using the equation \( \cos 3x = 0 \)

Given that \( \cos 3x = 0 \), we know that cosine is zero at odd multiples of \( \frac{\pi}{2} \). Therefore, we can write:

\[ 3x = \frac{\pi}{2} + k\pi \quad \text{(where \( k \) is an integer)} \]

Since \( x \) is acute, we will consider the smallest positive solution, which occurs when \( k = 0 \):

\[ 3x = \frac{\pi}{2} \]

Now, divide both sides by 3 to find \( x \):

\[ x = \frac{\pi}{6} \]

### Step 2: Calculate \( \cot^2 x \) and \( \csc^2 x \)

Now we need to find \( \cot^2 x \) and \( \csc^2 x \) for \( x = \frac{\pi}{6} \).

1. **Calculate \( \cot^2 x \)**:
\[ \cot x = \frac{\cos x}{\sin x} \]
For \( x = \frac{\pi}{6} \):
\[ \cot \frac{\pi}{6} = \frac{\cos \frac{\pi}{6}}{\sin \frac{\pi}{6}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]
Therefore,
\[ \cot^2 \frac{\pi}{6} = (\sqrt{3})^2 = 3 \]

2. **Calculate \( \csc^2 x \)**:
\[ \csc x = \frac{1}{\sin x} \]
For \( x = \frac{\pi}{6} \):
\[ \csc \frac{\pi}{6} = \frac{1}{\sin \frac{\pi}{6}} = \frac{1}{\frac{1}{2}} = 2 \]
Therefore,
\[ \csc^2 \frac{\pi}{6} = 2^2 = 4 \]

### Step 3: Find \( \cot^2 x - \csc^2 x \)

Now we can find the value of \( \cot^2 x - \csc^2 x \):

\[ \cot^2 x - \csc^2 x = 3 - 4 = -1 \]

### Final Answer

Thus, the value of \( \cot^2 x - \csc^2 x \) is:

\[ \boxed{-1} \]