Question

If x = b sec³θ and y = a tan³θ, prove that `(x/b)^(2/3) - (y/a)^(2/3) = 1`.

Answer 1

Given: x = b sec³θ, y = a tan³θ

To Prove: `(x/b)^(2/3) - (y/a)^(2/3) = 1`

Proof [Step-wise]:

1. Compute `(x/b)^(2/3)`:

`(x/b)^(2/3) = ((b sec^3θ)/b)^(2/3)` 

= `(sec^3θ)^(2/3)`

= sec²θ

2. Compute `(y/a)^(2/3)`:

`(y/a)^(2/3) = ((a tan^3θ)/a)^(2/3)` 

= `(tan^3θ)^(2/3)`

= tan²θ

3. Subtract the two results:

`(x/b)^(2/3) - (y/a)^(2/3) = sec^2θ - tan^2θ`

4. Use the fundamental trig identity sec²θ – tan²θ = 1 to obtain:

sec²θ – tan²θ = 1

Therefore `(x/b)^(2/3) - (y/a)^(2/3) = 1`.

Hence proved.