If `p=(secA - tanA) (secB - tanB) (secC - tanC)` = `(secA + tanA) (secB + tanB)(secC + tanC)`, then value of p is:

Options

A. `+-tan A tan B tanC`
B. `+-secA secB secC`
C. `+-1`
D. None of these

Answer

To solve the equation \( p = (secA - tanA)(secB - tanB)(secC - tanC) = (secA + tanA)(secB + tanB)(secC + tanC) \), we can follow these steps:

### Step 1: Set up the equation
Given:
\[ p = (secA - tanA)(secB - tanB)(secC - tanC) = (secA + tanA)(secB + tanB)(secC + tanC) \]

### Step 2: Multiply both sides
We can multiply both sides of the equation:
\[ p^2 = [(secA - tanA)(secA + tanA)][(secB - tanB)(secB + tanB)][(secC - tanC)(secC + tanC)] \]

### Step 3: Apply the difference of squares
Using the identity \( (a - b)(a + b) = a^2 - b^2 \):
\[ p^2 = (sec^2 A - tan^2 A)(sec^2 B - tan^2 B)(sec^2 C - tan^2 C) \]

### Step 4: Use the trigonometric identity
We know from trigonometric identities that:
\[ sec^2 \theta - tan^2 \theta = 1 \]
Thus, we can substitute this into our equation:
\[ p^2 = 1 \cdot 1 \cdot 1 = 1 \]

### Step 5: Solve for \( p \)
Taking the square root of both sides gives us:
\[ p = \pm 1 \]

### Conclusion
Thus, the value of \( p \) is:
\[ p = \pm 1 \]