If $$\cos \theta = \frac{2p}{p^2 + 1}$$, (p ≠ $$\pm 1$$) then $$\cosec \theta$$ is equal to:

$$\cos\theta=\frac{base}{hypotenuse}=\frac{2p}{p^2+1}$$

As per pythagoras theorem, $$\left(hypotenuse\right)^2\ =\left(base\right)^2\ +\ \left(height\right)^2$$

$$\left(p^2+1\right)^2\ =\left(2p\right)^2\ +\ \left(height\right)^2$$

$$p^4+2p^2+1\ =4p^2\ +\ \left(height\right)^2$$

$$p^4+2p^2+1\ -4p^2\ =\left(height\right)^2$$

$$p^4-2p^2+1\ =\left(height\right)^2$$

$$\left(p^2-1\right)^2\ =\left(height\right)^2$$

Hence $$\cosec \theta = \frac{hypotenuse}{height}$$

= $$\frac{p^2+1}{(p^2-1)}$$