$\frac{\sqrt{cosecx-1}}{\sqrt{cosecx+1}}$ is equal to:

secx - tanx

$\frac{\sqrt{cosecx-1}}{\sqrt{cosecx+1}}$

= $\frac{\sqrt{1/sinx-1}}{\sqrt{1/sinx+1}}$

= $\frac{\sqrt{1-sinx}}{\sqrt{1+sinx}}$

multiply and divide by \(\sqrt {1-sinx }\)

= $\frac{\sqrt{1-sinx}}{\sqrt{1+sinx}}$  × $\frac{\sqrt{1-sinx}}{\sqrt{1-sinx}}$

=  $\frac{\sqrt{(1-sinx)²}}{\sqrt{1-sin²x}}$

{ sin²θ  + cos²θ = 1 }

=   $\frac{\sqrt{(1-sinx)²}}{\sqrt{cos²x}}$

= \(\frac{1 -sinx }{cosx}\)

= secx - tanx