What is the simplified value of $$\sqrt{\frac{cosec\theta}{cosec\theta-1}+\frac{cosec\theta}{cosec\theta+1}}$$ ?

Expression : $$\sqrt{\frac{cosec\theta}{cosec\theta-1}+\frac{cosec\theta}{cosec\theta+1}}$$

= $$\sqrt{\frac{cosec\theta(cosec\theta+1)+cosec\theta(cosec\theta-1)}{(cosec\theta-1)(cosec\theta+1)}}$$

= $$\sqrt{\frac{(cosec^2\theta+cosec\theta)+(cosec^2\theta-cosec\theta)}{cosec^2\theta-1}}$$

Using, $$(cosec^2\theta-cot^2\theta=1)$$

= $$\sqrt{\frac{2cosec^2\theta}{cot^2\theta}}=\sqrt{2cosec^2\theta tan^2\theta}$$

= $$\sqrt{\frac{2}{sin^2\theta}\times\frac{sin^2\theta}{cos^2\theta}}$$

= $$\sqrt{\frac{2}{cos^2\theta}}=\sqrt{2sec^2\theta}$$

= $$\sqrt2sec\theta$$

=> Ans - (A)