Q240 / 270 · NDA 2016 (Phase 1) · Mathematics

If $A=\left(\cos 12^{\circ}-\cos 36^{\circ}\right)\left(\sin 96^{\circ}+\sin 24^{\circ}\right)$ and $\mathrm{B}=\left(\sin 60^{\circ}-\sin 12^{\circ}\right)\left(\cos 48^{\circ}-\cos 72^{\circ}\right)$, then what is $\frac{\mathrm{A}}{\mathrm{B}}$ equal to?

  • A
    –1
  • B
    0
  • Correct answer
    C
    1
  • D
    2

Solution

$\begin{aligned} & \text { Given } A=\left(\cos 12^{\circ}-\cos 36^{\circ}\right)\left(\sin 96^{\circ}+\sin 24^{\circ}\right) \\ & B=\left(\sin 60^{\circ}-\sin 12^{\circ}\right)\left(\cos 48^{\circ}-\cos 72^{\circ}\right) \\ & \frac{A}{B}=\frac{\left[2 \sin 24^{\circ} \sin 12^{\circ}\right]\left[2 \sin 60^{\circ} \cos 36^{\circ}\right]}{\left[2 \cos 36^{\circ} \sin 24^{\circ}\right]\left[2 \sin 60^{\circ} \sin 12^{\circ}\right]} \\ & \Rightarrow \frac{A}{B}=1\end{aligned}$

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

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