If `pcos e ctheta+qcottheta=2` and `p^2cos e c^2theta-q^2cot^2theta=5` then the value of `sqrt(81 p^(-2)- q^(-2))` is________

Explanation

To solve the problem, we are given two equations involving \( p \), \( q \), and \( \theta \):

1. \( p \csc \theta + q \cot \theta = 2 \) (Equation 1)
2. \( p^2 \csc^2 \theta - q^2 \cot^2 \theta = 5 \) (Equation 2)

We need to find the value of \( \sqrt{81 p^{-2} - q^{-2}} \).

### Step 1: Express \( \csc \theta \) and \( \cot \theta \)

From Equation 1, we can express \( p \csc \theta \) and \( q \cot \theta \):

\[ p \csc \theta = 2 - q \cot \theta \]

### Step 2: Substitute into Equation 2

Now, we can substitute \( p \csc \theta \) into Equation 2. We know:

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

Substituting this into Equation 2 gives:

\[ (2 - q \cot \theta)^2 - q^2 \cot^2 \theta = 5 \]

### Step 3: Expand the equation

Expanding the left-hand side:

\[ (2 - q \cot \theta)^2 = 4 - 4q \cot \theta + q^2 \cot^2 \theta \]

So, we have:

\[ 4 - 4q \cot \theta + q^2 \cot^2 \theta - q^2 \cot^2 \theta = 5 \]

This simplifies to:

\[ 4 - 4q \cot \theta = 5 \]

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

Rearranging gives:

\[ -4q \cot \theta = 5 - 4 \]
\[ -4q \cot \theta = 1 \]
\[ \cot \theta = -\frac{1}{4q} \]

### Step 5: Substitute back to find \( \csc \theta \)

Now substituting \( \cot \theta \) back into Equation 1:

\[ p \csc \theta + q \left(-\frac{1}{4q}\right) = 2 \]

This simplifies to:

\[ p \csc \theta - \frac{1}{4} = 2 \]
\[ p \csc \theta = 2 + \frac{1}{4} = \frac{9}{4} \]

### Step 6: Find \( \csc^2 \theta \)

Now we can find \( \csc^2 \theta \):

\[ \csc \theta = \frac{9}{4p} \]
\[ \csc^2 \theta = \left(\frac{9}{4p}\right)^2 = \frac{81}{16p^2} \]

### Step 7: Substitute \( \csc^2 \theta \) into Equation 2

Now substituting \( \csc^2 \theta \) into Equation 2:

\[ p^2 \left(\frac{81}{16p^2}\right) - q^2 \left(-\frac{1}{4q}\right)^2 = 5 \]
\[ \frac{81}{16} - \frac{1}{16} = 5 \]
\[ \frac{80}{16} = 5 \]

### Step 8: Find \( \sqrt{81 p^{-2} - q^{-2}} \)

Now we need to find \( \sqrt{81 p^{-2} - q^{-2}} \):

From the previous steps, we know:

\[ 81 p^{-2} = \frac{81}{p^2} \]
\[ q^{-2} = \frac{1}{q^2} \]

Thus:

\[ \sqrt{81 p^{-2} - q^{-2}} = \sqrt{16} = 4 \]

### Final Answer

The value is:

\[ \boxed{4} \]