If `psectheta-qtantheta=10&ptantheta-qsectheta=8`. Find `p^(2)-q^(2)+4`

Answer

To solve the problem, we have the equations:

1. \( p \sec \theta - q \tan \theta = 10 \)
2. \( p \tan \theta - q \sec \theta = 8 \)

We need to find \( p^2 - q^2 + 4 \).

### Step 1: Square the first equation
We start by squaring the first equation:

\[ (p \sec \theta - q \tan \theta)^2 = 10^2 \]

Expanding the left side using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):

\[ p^2 \sec^2 \theta - 2pq \sec \theta \tan \theta + q^2 \tan^2 \theta = 100 \]

### Step 2: Square the second equation
Now, we square the second equation:

\[ (p \tan \theta - q \sec \theta)^2 = 8^2 \]

Expanding this similarly:

\[ p^2 \tan^2 \theta - 2pq \tan \theta \sec \theta + q^2 \sec^2 \theta = 64 \]

### Step 3: Set up the equations
Now we have two equations:

1. \( p^2 \sec^2 \theta - 2pq \sec \theta \tan \theta + q^2 \tan^2 \theta = 100 \) (Equation 1)
2. \( p^2 \tan^2 \theta - 2pq \tan \theta \sec \theta + q^2 \sec^2 \theta = 64 \) (Equation 2)

### Step 4: Subtract Equation 2 from Equation 1
Now, we subtract Equation 2 from Equation 1:

\[ (p^2 \sec^2 \theta - p^2 \tan^2 \theta) + (q^2 \tan^2 \theta - q^2 \sec^2 \theta) - 2pq (\sec \theta \tan \theta - \tan \theta \sec \theta) = 100 - 64 \]

This simplifies to:

\[ p^2 (\sec^2 \theta - \tan^2 \theta) + q^2 (\tan^2 \theta - \sec^2 \theta) = 36 \]

### Step 5: Use the identity
Using the identity \( \sec^2 \theta - \tan^2 \theta = 1 \):

\[ p^2 (1) - q^2 (1) = 36 \]

Thus, we have:

\[ p^2 - q^2 = 36 \]

### Step 6: Find \( p^2 - q^2 + 4 \)
Now, we need to find \( p^2 - q^2 + 4 \):

\[ p^2 - q^2 + 4 = 36 + 4 = 40 \]

### Final Answer
The final answer is:

\[ \boxed{40} \]