COMEDK 2025 Evening Solved Paper

Concept:Given $\sin x + \sin^2 x = 1$, rewrite it as $\sin x = 1 - \sin^2 x = \cos^2 x$ and then express the required expression in terms of $\cos^2 x$ to simplify.Explanation:Start with $\sin x + \sin^2 x = 1$.This gives $\sin x = 1 - \sin^2 x = \cos^2 x$.Thus $\cos^2 x = \sin x$.Now consider the expression $\cos^8 x + 2\cos^6 x + \cos^4 x$.Factor it as $\cos^4 x(\cos^4 x + 2\cos^2 x + 1)$.The inner part is $(\cos^2 x + 1)^2$.So the expression equals $\cos^4 x (\cos^2 x + 1)^2$.Substitute $\cos^2 x = \sin x$ and $\cos^4 x = (\cos^2 x)^2 = \sin^2 x$.Then it becomes $\sin^2 x (\sin x + 1)^2$.From the original equation $\sin x + \sin^2 x = 1$, we have $\sin x = 1 - \sin^2 x = \cos^2 x$ (already used).Alternatively, $\sin^2 x = 1 - \sin x$.Replace $\sin^2 x$ with $1 - \sin x$ and $\sin x + 1$ remains as is.Now $\sin^2 x (\sin x + 1)^2 = (1 - \sin x)(\sin x + 1)^2$.Expand: $(1 - \sin x)(1 + 2\sin x + \sin^2 x) = (1 - \sin x)(1 + 2\sin x + \sin^2 x)$.But $\sin^2 x = 1 - \sin x$, so substitute: $(1 - \sin x)(1 + 2\sin x + 1 - \sin x) = (1 - \sin x)(2 + \sin x)$.Expand: $(1 - \sin x)(2 + \sin x) = 2 + \sin x - 2\sin x - \sin^2 x = 2 - \sin x - \sin^2 x$.Replace $\sin^2 x = 1 - \sin x$, then $2 - \sin x - (1 - \sin x) = 2 - \sin x - 1 + \sin x = 1$.Thus the required value is 1.Answer:$1$ (Option B)