If θ is an acute angle and sin θ cos θ = 2cos³ θ - 1/4 cos θ ,then the value of sin θ is:

If θ is an acute angle and sin θ cos θ = 2cos³ θ - 1/4 cos…

If $\theta$ is an acute angle and sin $\theta$ cos $\theta$ = 2cos³ $\theta$ - $\frac{1}{4}$ cos $\theta$ ,then the value of sin $\theta$ is:

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Correct answer: B

Explanation

The correct answer is option 2 i.e. $\sqrt{15}-1\over4$ .

sinθcosθ = 2cos³θ - (1/4)cosθ

⇒ sinθ = 2cos²θ - (1/4)

⇒ sinθ = 2(1 - sin²θ) - (1/4)

⇒ sinθ = 2 - 2sin²θ - (1/4)

⇒ sinθ = (7/4) - 2sin²θ

⇒ 4sinθ = 7 - 8sin²θ

⇒ 8sin²θ + 4sinθ - 7 = 0

⇒ sinθ = $\frac{-4\pm \sqrt {4^2-4 \times 8\times (-7)}}{2\times 8}$

⇒ sinθ = $\frac{-4\pm \sqrt {240}}{16}=\frac{-4\pm 4\sqrt {15}}{16}=\frac{-1\pm \sqrt {15}}{4}$

⇒ sinθ = (√15 - 1)/4

Therefore, the value of sinθ is (√15 - 1)/4.