Q62 / 160 · AP EAMCET 2019 (20 Apr Shift 2) · Mathematics

$\cos 66^{\circ}+\sin 84^{\circ}=$

  • A
    $\frac{1}{4}(\sqrt{3}+\sqrt{5})$
  • B
    $\frac{1}{4} \sqrt{5}(\sqrt{3}+1)$
  • C
    $\frac{1}{4}(\sqrt{3}+1)(\sqrt{5}+1)$
  • Correct answer
    D
    $\frac{1}{4} \sqrt{3}(\sqrt{5}+1)$

Solution

$$ \begin{aligned} & \cos 66^{\circ}+\sin 84^{\circ}=\cos 66^{\circ}+\cos 6^{\circ} \\ & =2 \cos 36^{\circ} \cos 30^{\circ} \\ & \quad\left[\because \cos C+\cos D=2 \cos \frac{C+D}{2} \cos \frac{C-D}{2}\right] \\ & =\sqrt{3} \cos 36^{\circ}=\sqrt{3} \frac{(\sqrt{5}+1)}{4} \quad\left[\because \cos 36^{\circ}=\frac{\sqrt{5}+1}{4}\right] \end{aligned} $$ Hence, option (d) is correct.

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

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