Step 1 :

Equation of the given line is

$x \sec \theta + y \: cosec \theta = a$

Slope of this line is

$ -\large\frac{\sec \theta}{cosec \theta } $$= -\large\frac{\sin \theta }{\cos \theta }$

Hence slope of the line perpendicular to the above line is $ \large\frac{\cos \theta}{\sin \theta}$

It passes through the point $( a \cos^3 \theta , a \sin^3 \theta )$

$ \therefore $ Equation of the required line is

$(y- a \sin^3 \theta ) = \large\frac{\cos \theta }{\sin \theta}$$ (x - a \cos^3 \theta )$

$ \Rightarrow \sin \theta (y- a \sin^3 \theta ) = \cos \theta (x - a \cos^3 \theta )$

$ \Rightarrow y \sin \theta - a \sin^4 \theta = x \cos \theta - a \cos^4 \theta $

$ \Rightarrow x \cos \theta - y \sin \theta = a \cos^4 \theta - a \sin^4 \theta $

$ \Rightarrow x \cos \theta - y \sin \theta = a ( cos^2 \theta + \sin^2 \theta ) ( \cos^2 \theta - \sin^2 \theta )$

But $ \cos^2 \theta + \sin^2 \theta = 1 $ and $ \cos^2 \theta - \sin^2 \theta = \cos 2\theta $

$ \therefore x \cos \theta - y \sin \theta = a \cos 2 \theta $

Hence the statement is true.