If `p and p` be the perpendicular form the origin upon the straight lines `x sec theta + y cosec theta = a and x cos theta -y sin theta = a cos 2 theta`
Prove that : `4p^2+p^2=a^2`

Explanation

To prove that \( 4p^2 + p'^2 = a^2 \), we will first find the expressions for \( p \) and \( p' \), which are the perpendicular distances from the origin to the given lines.

### Step 1: Find the perpendicular distance \( p \) from the origin to the line \( x \sec \theta + y \csc \theta = a \).

The formula for the perpendicular distance from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by:

\[
d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
...