If $ x=a \cos ^{3} \theta, y=b \sin ^{3} \theta $, prove that $ \left(\frac{x}{a}\right)^{2 / 3}+\left(\frac{y}{b}\right)^{2 / 3}=1 $

Given:

\( x=a \cos ^{3} \theta, y=b \sin ^{3} \theta \)

To do:

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

Solution:

We know that,

$\sin^2 A+\cos^2 A=1$

$\operatorname{cosec}^2 A-\cot^2 A=1$

$\sec^2 A-\tan^2 A=1$

$\cot A=\frac{\cos A}{\sin A}$

$\tan A=\frac{\sin A}{\cos A}$

$\operatorname{cosec} A=\frac{1}{\sin A}$

$\sec A=\frac{1}{\cos A}$

Therefore,

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

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

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

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

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

$=\cos ^{2} \theta+\sin ^{2} \theta$

$=1$

Hence proved.    

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Updated on: 10-Oct-2022

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