Q194 / 270 · NDA 2017 (Phase 1) · Mathematics

The expression $\frac{\sin \alpha+\sin \beta}{\cos \alpha+\cos \beta}$ is equal to

  • Correct answer
    A
    $\tan \left(\frac{\alpha+\beta}{2}\right)$
  • B
    $\tan \left(\frac{\alpha+\beta}{2}\right)$
  • C
    $\sin \left(\frac{\alpha+\beta}{2}\right)$
  • D
    $\cos \left(\frac{\alpha+\beta}{2}\right)$

Solution

$\begin{aligned} & \frac{\sin \alpha+\sin \beta}{\cos \alpha+\cos \beta}=\frac{2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)}{2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)} \\ & =\tan \left(\frac{\alpha+\beta}{2}\right)\end{aligned}$

Hence, the correct answer is option: 'A'.

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