If `("cosec" theta +cot theta)=x` find `"cosec" theta - cot theta`.

Answer

To solve the problem, we start with the equation given:

\[ \csc \theta + \cot \theta = x \]

We need to find the value of:

\[ \csc \theta - \cot \theta \]

### Step 1: Use the identity for \(\csc^2 \theta\) and \(\cot^2 \theta\)

We know from trigonometric identities that:

\[ \csc^2 \theta = 1 + \cot^2 \theta \]

### Step 2: Rewrite \(\csc \theta - \cot \theta\) in terms of \(\csc \theta + \cot \theta\)

We can express \(\csc \theta - \cot \theta\) using the identity:

\[ (\csc \theta - \cot \theta)(\csc \theta + \cot \theta) = \csc^2 \theta - \cot^2 \theta \]

This can be factored as:

\[ \csc^2 \theta - \cot^2 \theta = 1 \]

### Step 3: Substitute the value of \(\csc \theta + \cot \theta\)

Now we substitute \(x\) for \(\csc \theta + \cot \theta\):

\[ (\csc \theta - \cot \theta)(x) = 1 \]

### Step 4: Solve for \(\csc \theta - \cot \theta\)

To find \(\csc \theta - \cot \theta\), we rearrange the equation:

\[ \csc \theta - \cot \theta = \frac{1}{x} \]

### Final Answer

Thus, we have:

\[ \csc \theta - \cot \theta = \frac{1}{x} \]

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