Question

Prove the following identities:

`("cosec"  A - sin A)/("cosec"  A + sin A) = (sec^2A - tan^2A)/(sec^2A + tan^2A)`

Answer 1

LHS = `{{1/(sin A) - sin A}}/{{1/(sin A) + sin A}}`

= `((1 - sin^2A))/((1 + sin^2A))`

= `(cos^2A)/((1 + sin^2A))`

RHS = `1/((1/(cos^2A) + (sin^2A)/(cos^2A))`   ...[∵ sec²A – tan²A = 1]

= `(cos^2A)/((1 + sin^2A))`

∴ LHS = RHS.