Question

Prove that (sec²θ−1)(cosec²θ−1) = 1

Answer 1

Identities : sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1

LHS = ( sec²θ - 1)(cosec²θ − 1)

= tan²θ x cot²θ

= 1.......(∵ tanθ = \(\frac{1}{cot\ θ}\))

= RHS

Hence proved.

Answer 2

Using Trigonometric Identities

1+tan² θ  = sec² θ

1+cot² θ  = cosec² θ

Now L.H.S

(1+tan² θ -1)(1+cot² θ-1)

(tan² θ)(cot² θ)

tan² θ X 1/ tan² θ  (because cotθ = 1/tanθ)

=1