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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NTA Abhyas 2020 · Mathematics · Trigonometric Ratios & Identities

If 2β is the geometric mean between sin α and cos α, then 4β is...

If sin⁡2β\sin 2\beta is the geometric mean between sinαsin \alpha and cosα,cos \alpha, then cos⁡4β\cos 4\beta is equal to

Options

  1. A2sin⁡2(π4−α)2\sin^{2} \left(\dfrac{\pi}{4} - \alpha\right)
  2. B2cos⁡2(π4−α)2\cos^{2} \left(\dfrac{\pi}{4} - \alpha\right)
  3. C2cos⁡2(π2+α)2\cos^{2} \left(\dfrac{\pi}{2} + \alpha\right)
  4. D2sin⁡2(π4+α)2\sin^{2} \left(\dfrac{\pi}{4} + \alpha\right)

Correct answer

A. 2sin⁡2(π4−α)2\sin^{2} \left(\dfrac{\pi}{4} - \alpha\right)

Step-by-step solution

sin⁡22β=sinαcosα\sin^{2} 2\beta = sin \alpha cos \alpha
cos⁡4β=1−2sin⁡22β\cos 4\beta = 1 - 2\sin^{2} 2\beta
=sin⁡2α+cos⁡2α−2sinαcosα= \sin^{2} \alpha + \cos^{2} \alpha - 2sin \alpha cos \alpha
=(sinα−cosα)2= {\left(sin \alpha - cos \alpha\right)}^{2}
=(2(sinα⋅12−cosα12))2= \left(\sqrt{2}\left(sin\alpha \cdot \dfrac{1}{2} - cos\alpha\dfrac{1}{2}\right)\right)^{2}
=2(sinαcosπ4−cosαsinπ4)2= 2{\left(sin \alpha cos \dfrac{\pi}{4} - cos \alpha sin \dfrac{\pi}{4}\right)}^{2}
=2(sin⁡(α−π4))= 2{\left(\sin \left(\alpha - \dfrac{\pi}{4}\right)\right)}^{2} = 2sin⁡2\sin^{2} \left(\dfrac{\pi}{4} -\alpha\right)
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