Question

Prove the following:

`1/(1 + sinA) - 1/(1 - sinA) = - 2secA tanA`

Answer 1

LHS = `((1 - sinA) - (1 + sinA))/((1 + sinA)(1 - sinA))`

`(1 - sinA - 1 - sinA)/((1 + sinA)(1 - sinA))`

= `(-2sinA)/((1 + sinA)(1 - sinA))`

Simplify the denominator:

`(-2sinA)/(1 - sin^2A)`

1 − sin² A = cos² A

∴ `(-2sinA)/(cos^2A)`

Separate the terms to match the RHS = cos² A as cos A.cos A

`-2. 1/cosA.sinA/cosA`

Substitute the trigonometric definitions:

`1/cosA = secA and sinA/cosA = tanA`

− sec A . tan A

LHS = RHS