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BITSAT 2014
Exam
BITSAT
Year
2014
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

BITSAT 2014 · Mathematics · Trigonometric Ratios & Identities

^ 2 ((π)/(6)+θ)- ^ 2 ((π)/(6)-θ)=

cos⁡2\cos ^{2}\left(\frac{\pi}{6}+\theta\right)-\sin ^{2}(π6−θ)=\left(\frac{\pi}{6}-\theta\right)=

Options

  1. A12cos⁡2θ\frac{1}{2} \cos 2 \theta
  2. B0
  3. C−12cos⁡2θ-\frac{1}{2} \cos 2 \theta
  4. D12\frac{1}{2}

Correct answer

A. 12cos⁡2θ\frac{1}{2} \cos 2 \theta

Step-by-step solution

cos⁡2(π6+θ)−sin⁡2(π6−θ)=cos⁡(π6+θ+π6−θ)cos⁡(π6+θ−π6+θ)=cos⁡2π6cos⁡2θ=12cos⁡2θ\begin{aligned} & \cos ^{2}\left(\frac{\pi}{6}+\theta\right)-\sin ^{2}\left(\frac{\pi}{6}-\theta\right) \\ &=\cos \left(\frac{\pi}{6}+\theta+\frac{\pi}{6}-\theta\right) \cos \left(\frac{\pi}{6}+\theta-\frac{\pi}{6}+\theta\right) \\ &=\cos \frac{2 \pi}{6} \cos 2 \theta=\frac{1}{2} \cos 2 \theta \end{aligned}
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