If \cot ^{2} \theta+\cot ^{4} \theta=2, then the value of 2 \sin ^{4} \theta+\sin ^{2} \theta is

SSC CHSL 11 Aug 2021 Shift 2 Paper

Section: Quantitative Aptitude

Question : 74 of 100

Marks: +1, -0

\cot ^{2} θ+\cot ^{4} θ=2

, then the value of

2 \sin ^{4} θ+\sin ^{2} θ

Solution:

Given,

\cot ^{2} θ+\cot ^{4} θ=2

⇒ \cot ^{2} θ(1+\cot ^{2} θ)=2

⇒ \;\; 1+\cot ^{2} θ=\;{2}/{\cot ^{2} θ}

⇒ \;\; cosec^{2} θ × \cot ^{2} θ=2

⇒ \;\; \;{1}/{\sin ^{2} θ} × \;{\cos ^{2} θ}/{\sin ^{2} θ}=2

⇒ \;\; \cos ^{2} θ=2 \sin ^{4} θ

⇒ \;\; (1-\sin ^{2} θ)=2 \sin ^{4} θ

⇒ \;\; 2 \sin ^{4} θ+\sin ^{2} θ=1

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