Question

If (sec A – tan A) = x then prove that `(1 + x^2)/(1 - x^2)` = cosec A.

Answer 1

Given: (sec A – tan A) = x

To Prove: `(1 + x^2)/(1 - x^2)` = cosec A

Proof [Step-wise]:

1. Use the standard identity (sec A + tan A)(sec A – tan A) = sec²A – tan²A = 1.
Since sec A – tan A = x, we get `sec A + tan A = 1/x`.

2. Write sec A ± tan A in terms of sin A and cos A:

`sec A ± tan A = (1 ± sin A)/(cos A)`.

So `(1 + sin A)/(cos A) = 1/x` and `(1 - sin A)/(cos A) = x`.

3. Add the two equations from step 2:

`(1 + sin A)/(cos A) + (1 - sin A)/(cos A) = 1/x + x`

⇒ `2/(cos A) = 1/x + x`

⇒ `cos A = 2/(x + 1/x)`

⇒ `cos A = (2x)/(1 + x^2)`

4. Substitute cos A into `(1 - sin A)/cos A = x` to find sin A:

1 – sin A = x

`cos A = x xx (2x)/(1 + x^2)`

= `(2x^2)/(1 + x^2)` 

⇒ `sin A = 1 - (2x^2)/(1 + x^2)`

= `(1 + x^2 − 2x^2)/(1 + x^2)` 

⇒ `sin A = (1 - x^2)/(1 + x^2)`

5. Therefore `"cosec"  A = 1/(sin A) = (1 + x^2)/(1 - x^2)`.

`(1 + x^2)/(1 - x^2)` = cosec A, as required.