Question

Prove the following trigonometric identities.

if x = a cos^3 theta, y = b sin^3 theta` " prove that " `(x/a)^(2/3) + (y/b)^(2/3) = 1`

Answer 1

Given:

`x = a cos^3 theta`

`=> x/a = cos^3 theta`

`x = b sin^3 theta`

`=> y/b = sin^3 theta`

We have to prove `(x/a)^(2/3) + (y/b)^(2/3) = 1`

We know that `sin^2 theta + cos^2 theta =1`

So we have

`(x/a)^(2/3) + (yb)^(2/3) = (cos^2 theta)^(2/3) + (sin^3 theta)^(2/3)`

`=> (x/a)^(2/3) + (y/b)^(2/3) = cos^2 theta + sin^2 theta` 

`=> (x/a)^(2/3) + (y/b)^(2/3) = 1`

Hence proved