Q62 / 160 · AP EAMCET 2023 (17 May Shift 1) · Mathematics

$\cos 12^{\circ} \cdot \cos 24^{\circ} \cdot \cos 36^{\circ} \cdot \cos 48^{\circ} \cdot \cos 72^{\circ} \cdot \cos 84^{\circ}=$

  • A
    $\frac{1}{32}$
  • B
    $\frac{1}{16}$
  • Correct answer
    C
    $\frac{1}{64}$
  • D
    $\frac{1}{128}$

Solution

$\begin{aligned} & \text {} \cos 12 \cos 24 \cos 36 \cos 48 \cos 72 \cos 84 \\ & =\frac{2 \sin 24 \cos 24 \cos 36 \cos 48 \cos 72 \cos 84}{4 \sin 12} \\ & =\frac{-2 \sin 48 \cos 48 \cos 36 \cos 72 \cos (180-84)}{8 \sin 12}\end{aligned}$ $\begin{aligned} & =\frac{-2 \sin 96 \cos 96 \cos 36 \cos 72}{16 \sin 12} \\ & =\frac{\sin 192 \cos 36 \cos 72}{16 \sin 12} \\ & =\frac{\sin 12(2 \sin 36 \cos 36) \cos 72}{32 \sin 12 \sin 36} \\ & =\frac{2 \sin 72 \cos 72}{64 \sin 36}=\frac{\sin 144}{64 \sin 36}=\frac{\sin (180-144)}{64 \sin 36}=\frac{1}{64}\end{aligned}$

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

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