If tanθ = 1/√11 0 < θ < π/2, then the value of $$\frac{cosec^{2}\theta-\sec^2\theta}{cosec^2\theta+\sec^2\theta}$$

Expression : $$tan\theta = \frac{1}{\sqrt{11}}$$

We know that, $$sec\theta = \sqrt{1 + tan^2 \theta}$$

=> $$sec\theta = \sqrt{1 + \frac{1}{11}} = \sqrt{\frac{12}{11}}$$

Now, $$cosec\theta = \frac{sec\theta}{tan\theta}$$

=> $$cosec\theta = \sqrt{12}$$

To find : $$\frac{cosec^{2}\theta-\sec^2\theta}{cosec^2\theta+\sec^2\theta}$$

= $$\frac{12 - \frac{12}{11}}{12 + \frac{12}{11}}$$

= $$\frac{1 - \frac{1}{11}}{1 + \frac{1}{11}}$$

= $$\frac{10}{12} = \frac{5}{6}$$