If sin θ + $$sin^{2}$$ θ = 1, then what is the value of $$(cos^{12}\ \theta\ + 3\ cos^{10}\ \theta+3\ cos^{8}\ \theta\ + cos^{6}\ \theta -1)$$

Given $$sin \theta + sin^{2} \theta = 1$$

$$\Rightarrow sin$$ θ = 1 - $$sin^{2}$$ θ $$\Rightarrow$$ $$sin$$ θ = $$cos^{2}$$ θ .....(1)

Now, $$(cos^{12}\ \theta\ + 3\ cos^{10}\ \theta+3\ cos^{8}\ \theta\ + cos^{6}\ \theta -1)$$

$$(cos^{4} \theta$$ + $$cos^{2} \theta)^{3}$$ - 1

Substitute equation (1) in the above equation

$$(sin^{2} \theta$$ + $$cos^{2} \theta)^{2}$$ - 1

1 - 1 = 0

Hence, option B is the correct answer.