Prove the following trigonometric identities.

Given:

$x=a \cos ^{3} \theta$

$\Rightarrow \frac{x}{a}=\cos ^{3} \theta$

$x=b \sin ^{3} \theta$

$\Rightarrow \frac{y}{b}=\sin ^{3} \theta$

We have to prove $\left(\frac{x}{a}\right)^{2 / 3}+\left(\frac{y}{b}\right)^{2 / 3}=1$

We know that $\sin ^{2} \theta+\cos ^{2} \theta=1$

So, we have

$\left(\frac{x}{a}\right)^{2 / 3}+\left(\frac{y}{b}\right)^{2 / 3}=\left(\cos ^{3} \theta\right)^{2 / 3}+\left(\sin ^{3} \theta\right)^{2 / 3}$

$\Rightarrow \quad\left(\frac{x}{a}\right)^{2 / 3}+\left(\frac{y}{b}\right)^{2 / 3}=\cos ^{2} \theta+\sin ^{2} \theta$

$\Rightarrow \quad\left(\frac{x}{a}\right)^{2 / 3}+\left(\frac{y}{b}\right)^{2 / 3}=1$

Hence proved.