If $$\frac{1}{cos\theta}-\frac{1}{cot\theta}=\frac{1}{p}$$, then what is the value of cos θ?

Expression : $$\frac{1}{cos\theta}-\frac{1}{cot\theta}=\frac{1}{p}$$

=> $$\frac{1}{cos\theta}-\frac{sin\theta}{cos\theta}=\frac{1}{p}$$

=> $$\frac{1-\sqrt{1-cos^2\theta}}{cos\theta}=\frac{1}{p}$$

Let $$cos\theta=x$$

=> $$1-\sqrt{1-x^2} = \frac{x}{p}$$

=> $$1-\frac{x}{p}=\sqrt{1-x^2}$$

Squaring both sides, we get :

=> $$(1-\frac{x}{p})^2=(\sqrt{1-x^2})^2$$

=> $$1+\frac{x^2}{p^2}-2\frac{x}{p}=1-x^2$$

=> $$\frac{x^2}{p^2}-2\frac{x}{p}+x^2=0$$

=> $$\frac{x^2-2xp+x^2p^2}{p^2}=0$$

=> $$x-2p+xp^2=0$$

=> $$x(1+p^2)=2p$$

=> $$x=cos\theta=\frac{2p}{1+p^2}$$

=> Ans - (D)