UPSC CDS 2 2019 Math Paper

Concept:Use the given trigonometric identity to express $\cos x$ in terms of $\sin^2 x$, then rewrite the polynomial in $\sin x$ as a perfect cube.Explanation:Given: $\cos^{2}x + \cos x = 1$.Rearrange: $\cos x = 1 - \cos^{2}x = \sin^{2}x$.So $\sin^{2}x = \cos x$ and $\sin^{4}x = \cos^{2}x$.Now consider the expression: $\sin^{12}x + 3\sin^{10}x + 3\sin^{8}x + \sin^{6}x$.Recognise it as $(\sin^{4}x)^{3} + 3(\sin^{4}x)^{2}(\sin^{2}x) + 3(\sin^{4}x)(\sin^{2}x)^{2} + (\sin^{2}x)^{3}$.That is the expansion of $(\sin^{4}x + \sin^{2}x)^{3}$.Substitute $\sin^{2}x = \cos x$ and $\sin^{4}x = \cos^{2}x$:$(\cos^{2}x + \cos x)^{3}$.But from the given condition, $\cos^{2}x + \cos x = 1$.Therefore the expression equals $1^{3} = 1$.Answer:The value is $1$, which corresponds to option A.