If $$0^0<\theta<90^0$$ and $$Cosec\theta=Cot^2\theta$$, then the value of the expression $$Cosec^4\theta-2Cosec^3\theta+Cot^2\theta$$ is equal to:

Given : $$cosec\theta=cot^2\theta$$

=> $$cosec\theta=cosec^2\theta-1$$

=> $$cosec^2\theta-cosec\theta-1=0$$ -------------(i)

Squaring both sides, we get :

=> $$cosec^4\theta=(cosec\theta+1)^2$$

=> $$cosec^4\theta=cosec^2\theta+2cosec\theta+1$$ ----------------(ii)

To find : $$cosec^4\theta-2cosec^3\theta+cot^2\theta$$

= $$(cosec^2\theta+2cosec\theta+1)-2cosec^3\theta+(cosec^2\theta-1)$$     [Using equation (ii)]

= $$-2cosec^3\theta+2cosec^2\theta+2cosec\theta$$

= $$-2cosec\theta(cosec^2\theta-cosec\theta-1)$$

Substituting value from equation (i),

= $$-2cosec\theta\times(0)=0$$

=> Ans - (B)