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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2017 · Mathematics · Trigonometric Ratios & Identities

The expression ( α+ β)/( α+ β) is equal to

The expression sin⁡α+sin⁡βcos⁡α+cos⁡βisequalto\frac{\sin \alpha+\sin \beta}{\cos \alpha+\cos \beta} is equal to

Options

  1. Atan⁡(α+β2)\tan \left(\frac{\alpha+\beta}{2}\right)
  2. Btan⁡(α+β2)\tan \left(\frac{\alpha+\beta}{2}\right)
  3. Csin⁡(α+β2)\sin \left(\frac{\alpha+\beta}{2}\right)
  4. Dcos⁡(α+β2)\cos \left(\frac{\alpha+\beta}{2}\right)

Correct answer

A. tan⁡(α+β2)\tan \left(\frac{\alpha+\beta}{2}\right)

Step-by-step solution

sin⁡α+sin⁡βcos⁡α+cos⁡β=2sin⁡(α+β2)cos⁡(α−β2)2cos⁡(α+β2)cos⁡(α−β2)=tan⁡(α+β2)\begin{aligned} & \frac{\sin \alpha+\sin \beta}{\cos \alpha+\cos \beta}=\frac{2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)}{2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)} \\ & =\tan \left(\frac{\alpha+\beta}{2}\right)\end{aligned}
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