Given : `4 sintheta = 3 cos theta` , find the value of:
`cot^2 theta - "cosec"^2 theta`

Answer

To solve the problem given \( 4 \sin \theta = 3 \cos \theta \) and find the value of \( \cot^2 \theta - \csc^2 \theta \), we can follow these steps:

### Step 1: Express cotangent and cosecant in terms of sine and cosine.
We know that:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]
and
\[ \csc \theta = \frac{1}{\sin \theta} \]
Thus, we can express \( \cot^2 \theta \) and \( \csc^2 \theta \):
\[ \cot^2 \theta = \left(\frac{\cos \theta}{\sin \theta}\right)^2 = \frac{\cos^2 \theta}{\sin^2 \theta} \]
\[ \csc^2 \theta = \left(\frac{1}{\sin \theta}\right)^2 = \frac{1}{\sin^2 \theta} \]

### Step 2: Substitute into the expression \( \cot^2 \theta - \csc^2 \theta \).
Now, substituting these into the expression:
\[ \cot^2 \theta - \csc^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} - \frac{1}{\sin^2 \theta} \]
This can be combined using a common denominator:
\[ = \frac{\cos^2 \theta - 1}{\sin^2 \theta} \]

### Step 3: Use the Pythagorean identity.
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
This implies:
\[ \cos^2 \theta = 1 - \sin^2 \theta \]
Thus, we can rewrite \( \cos^2 \theta - 1 \):
\[ \cos^2 \theta - 1 = (1 - \sin^2 \theta) - 1 = -\sin^2 \theta \]

### Step 4: Substitute back into the expression.
Now substituting this back into our expression:
\[ \cot^2 \theta - \csc^2 \theta = \frac{-\sin^2 \theta}{\sin^2 \theta} = -1 \]

### Final Answer:
Thus, the value of \( \cot^2 \theta - \csc^2 \theta \) is:
\[ \boxed{-1} \]