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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2022 · Mathematics · Trigonometric Ratios & Identities

What is ^2 ((π)/(4)+θ)- ^^2( (π)/(4)- θ) equal to?

What is sin⁡2\sin ^2\left(\frac{\pi}{4}+\theta\right)-\sin ^^2\left(π4−\frac{\pi}{4}-\theta\right) equal to?

Options

  1. Asin⁡2θ\sin 2 \theta
  2. Bcos⁡2θ\cos 2 \theta
  3. C2sin⁡θ2 \sin \theta
  4. D2cos⁡θ2 \cos \theta

Correct answer

A. sin⁡2θ\sin 2 \theta

Step-by-step solution

sin⁡2(π4+θ)−sin⁡2(π4−θ)=sin⁡[π4+θ+π4−θ]sin⁡[π4+θ−π4+θ]\begin{aligned} & \sin ^2\left(\frac{\pi}{4}+\theta\right)-\sin ^2\left(\frac{\pi}{4}-\theta\right) \\ & =\sin \left[\frac{\pi}{4}+\theta+\frac{\pi}{4}-\theta\right] \sin \left[\frac{\pi}{4}+\theta-\frac{\pi}{4}+\theta\right]\end{aligned}\left[\because \quad \sin ^2 x-\sin ^2 y=\sin (x+y) \cdot \sin (x-y)\right]=\sin \frac{\pi}{2} \cdot \sin 2 \theta=\sin 2 \theta$
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