If `p` and `q` are the lengths of perpendicular from the origin to the line `xcos(theta)-ysin(theta)=kcos(2theta)` and` xsec(theta)+ycosec(theta)=k` respectively , then prove that `p^2+4q^2=k^2`

Explanation

To prove that \( p^2 + 4q^2 = k^2 \), we will calculate the lengths of the perpendiculars \( p \) and \( q \) from the origin to the given lines and then derive the required relationship.

### Step 1: Find \( p \)

The equation of the first line is given by:

\[
x \cos \theta - y \sin \theta = k \cos 2\theta
...