If tan θ + cot θ = x, then what is the value of $$tan^4θ + cot^4θ$$ ?

Given : $$tan\theta+cot\theta=x$$

Squaring both sides,

=> $$(tan\theta+cot\theta)^2=(x)^2$$

=> $$tan^2\theta+cot^2\theta+2(tan\theta)(cot\theta)=x^2$$

=> $$tan^2\theta+cot^2\theta+2=x^2$$     [$$\because tan\theta cot\theta=1$$]

=> $$tan^2\theta+cot^2\theta=x^2-2$$

Again squaring both sides, we get :

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

=> $$tan^4\theta+cot^4\theta+2(tan^2\theta)(cot^2\theta)=x^4-4x^2+4$$

=> $$tan^4\theta+cot^4\theta+2=x^4-4x^2+4$$

=> $$tan^4\theta+cot^4\theta=x^2(x^2-4)+4-2$$

=> $$tan^4\theta+cot^4\theta=x^2(x^2-4)+2$$

=> Ans - (D)