If $$tan^2θ + cot^2θ = 2$$, then what is the value of 2secθ - cosec θ?

Given : $$tan^2θ + cot^2θ = 2$$

=> $$tan^2θ + cot^2θ - 2=0$$

=> $$tan^2\theta+cot^2\theta-2(tan\theta)(cot\theta)=0$$     [$$\because tan\theta cot\theta=1$$]

=> $$(tan\theta-cot\theta)^2=0$$

=> $$tan\theta-cot\theta=0$$

=> $$tan\theta=cot\theta$$

=> $$tan\theta=tan(90^\circ-\theta)$$

=> $$\theta=90^\circ-\theta$$

=> $$\theta+\theta=2\theta=90^\circ$$

=> $$\theta=\frac{90}{2}=45^\circ$$

$$\therefore$$ $$2sec\theta-cosec\theta$$

= $$2sec(45^\circ)-cosec(45^\circ)$$

= $$2\sqrt2-\sqrt2=\sqrt2$$

=> Ans - (D)