Q65 / 160 · TS EAMCET 2023 (13 May Shift 2) · Mathematics
$$
\sin 6^{\circ}+\sin 54^{\circ}+\sin 126^{\circ}+\cos 156^{\circ}=
$$
- Correct answerA$\frac{\sqrt{5}+1}{4}$
- B$\frac{\sqrt{5}-1}{4}$
- C$-\frac{1}{2}$
- D$\frac{3}{4}$
Solution
$\begin{aligned} & \text { } \sin 6^{\circ}+\sin 54^{\circ}+\sin 126^{\circ}+\cos 156^{\circ} \\ & =\sin 6^{\circ}+\cos \left(90^{\circ}+66^{\circ}\right)+\sin 54^{\circ}+\sin \left(90^{\circ}+36^{\circ}\right) \\ & =\sin 6^{\circ}-\sin 66^{\circ}+\sin 54^{\circ}+\cos 36^{\circ} \\ & =-2 \cos 36^{\circ} \sin 30^{\circ}+2 \cos 36^{\circ} \\ & =-\cos 36^{\circ}+2 \cos 36^{\circ} \\ & =\cos 36^{\circ}=\frac{\sqrt{5}+1}{4}\end{aligned}$
Hence, the correct answer is option: 'A'.