$\frac{{cosec} \theta}{{cosec} \theta-1}+\frac{{cosec} \theta}{{cosec} \theta+1}-\tan ^2 \theta, 0^{\circ}<\theta<90^{\circ},$ is equal to:

$\sec ^2 \theta+1$

We are  given, 

\(\frac{cosecθ}{cosecθ-1 }\) + \(\frac{cosecθ}{cosecθ+1 }\) - tan²θ

= \(\frac{cosecθ(cosecθ+1) +cosecθ(cosecθ-1)  }{cosec² θ - 1² }\)  - tan²θ

 = \(\frac{ 2 cosec² θ }{cosec² θ - 1² }\)  - tan²θ

{ using , cosec²θ - cot ²θ = 1 }

= \(\frac{ 2 cosec² θ }{cot ² θ}\)  - tan²θ

= \(\frac{ 2 }{cos ² θ}\)  - tan²θ

= 2 sec²θ - tan²θ

= sec²θ + sec²θ - tan² θ

{ using , sec²θ - tan² θ  = 1 }

= sec²θ + 1