The second order derivative of $a \sin ^3 t$ with respect to a $\cos ^3 t$ at $t=\pi / 4$, is

$\frac{4 \sqrt{2}}{3 a}$

Let $y=a \sin ^3 t$ and $x=a \cos ^3 t$. Then,

$\frac{d y}{d t}=3 a \sin ^2 t \cos t$  and  $\frac{d x}{d t}=-3 a \cos ^2 t \sin t$

$\Rightarrow \frac{d y}{d x}=-\tan t$

$\Rightarrow \frac{d^2 y}{d x^2}=-\sec ^2 t \frac{d t}{d x}=\frac{\sec ^2 t}{3 a \cos ^2 t \sin t}$

$\Rightarrow \left(\frac{d^2 y}{d x^2}\right)_{t=\pi / 4}=\frac{4 \sqrt{2}}{3 a}$