If p and q are the lengths of perpendiculars from the origin to the lines `xcostheta-ysintheta=kcos2theta`and `xsectheta+yc o s e ctheta=k`, respectively, prove that `p^2+4q^2=k^2`.

Explanation

To prove that \( p^2 + 4q^2 = k^2 \), where \( p \) and \( q \) are the lengths of the perpendiculars from the origin to the given lines, we will follow these steps:

### Step 1: Identify the lines and their equations
The two lines given in the question are:
1. \( x \cos \theta - y \sin \theta = k \cos 2\theta \)
2. \( x \sec \theta + y \cos \sec \theta = k \)

### Step 2: Calculate the length of the perpendicular \( p \) from the origin to the first line
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