Question

For what values of `x` does `sqrt((1-sinx)/(1+sinx))=secx-tanx` hold?

Answer 1

Identity holds for all values of `x`, irrespective of quadrant.

Explanation:

`sqrt((1-sinx)/(1+sinx))`

= `sqrt(((1-sinx)^2)/((1+sinx)(1-sinx))`

= `sqrt((1-sinx)^2/(1-sin^2x))`

= `sqrt((1-sinx)^2/cos^2x)`

= `(1-sinx)/cosx`

= `1/cosx-sinx/cosx`

= `secx-tanx`

Above is an identity and hence it holds for all values of `x` i.e. for any quadrant.