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 lines given by the equations \( x \cos \theta - y \sin \theta = k \cos 2\theta \) and \( x \sec \theta + y \csc \theta = k \) respectively, we can follow these steps:

### Step 1: Rewrite the equations of the lines
The first line can be rewritten as:
\[ \frac{x}{\cos \theta} - \frac{y}{\sin \theta} = k \cos 2\theta \]
The second line can be rewritten as:
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