Q175 / 270 · NDA 2020 · Mathematics

If $\sin x+\sin y=\cos y-\cos x$, where $0 \lt y \lt x \lt \frac{\pi}{2}$, then what is $\tan \left(\frac{x-y}{2}\right)$ equal to?

  • A
    0
  • B
    $\frac{1}{2}$
  • Correct answer
    C
    1
  • D
    2

Solution

$\sin x+\sin y=\cos y-\cos x$ $\frac{\sin x+\sin y}{\cos y-\cos x}=1$ $\frac{2 \sin \left(\frac{x+y}{2}\right) \cdot \cos \left(\frac{x-y}{2}\right)}{2 \sin \left(\frac{x+y}{2}\right) \cdot \sin \left(\frac{x-y}{2}\right)}=1$ $\cot \left(\frac{x-y}{2}\right)=1 \Rightarrow \tan \left(\frac{x-y}{2}\right)=1$

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

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