Find the radius of curvature at any point $(r, \theta)$ for the curve $a^2 = r^2\cos (2\theta)$
My Attempt: $$a^2 = r^2 \cos (2\theta)$$ $$r^2 = a^2 \sec (2\theta)$$ Differentiating both sides $$2rr_{1} = 2a^2 sec(2\theta)\cdot \tan (2\theta)$$ $$rr_{1} = a^2 \sec (2\theta) \tan (2\theta)$$
In polar coordinates, the radius of curvature is given by (have a look here)$$\large R=\frac{\left(r^2+\left(\frac{dr}{d\theta}\right)^2\right)^{3/2}}{\left|r^2+2\left(\frac{dr}{d\theta}\right)^2-r\left(\frac{d^2r}{d\theta^2}\right)\right|}$$