If \(\cos^2 x+\cos^4 x=1\), where \(0<x<\dfrac{\pi}{2}\), then what is \((\sin^2 x+\sin^3 x)+(\sin^3 x+\sin^4 x)\) equal to?
1
Let \(s=\sin x\). Writing \(\cos^2 x=1-s^2\), the relation \(\cos^2x+\cos^4x=1\) becomes \(s+s^2=1\), i.e. \(\cos^2 x=s\). From \(s^2=1-s\) we get \(s^3=s-s^2=2s-1\) and \(s^4=2s^2-s=2-3s\). So \((\sin^2x+\sin^3x)+(\sin^3x+\sin^4x)=s^2+2s^3+s^4=(1-s)+2(2s-1)+(2-3s)=1\). The correct option is (d).
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