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Trigonometry – II — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometry – II4 Marks
Question
$\cos 12^{\circ}+\cos 84^{\circ}+\cos 156^{\circ}+\cos 132^{\circ}=-\frac{1}{2}$
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Answer

L.H.S.
$\begin{aligned}
= & \cos 12^{\circ}+\cos 84^{\circ}+\cos 156^{\circ}+\cos 132^{\circ} \\
= & \left(\cos 132^{\circ}+\cos 12^{\circ}\right)+\left(\cos 156^{\circ}+\cos 84^{\circ}\right) \\
= & 2 \cos \left(\frac{132^{\circ}+12^{\circ}}{2}\right) \cos \left(\frac{132^{\circ}-12^{\circ}}{2}\right) \\
& +2 \cos \left(\frac{156^{\circ}+84^{\circ}}{2}\right) \cos \left(\frac{156^{\circ}-84^{\circ}}{2}\right) \\
= & 2 \cos 72^{\circ} \cos 60^{\circ}+2 \cos 120^{\circ} \cos 36^{\circ} \\
= & 2 \cos 72^{\circ} \cos 60^{\circ}+2 \cos \left(180^{\circ}-60^{\circ}\right) \cos 36^{\circ} \\
= & 2 \cos 72^{\circ} \cos 60^{\circ}+2\left(-\cos 60^{\circ}\right) \cos 36^{\circ} \\
= &2 \cos 72^{\circ}\left(\frac{1}{2}\right)-2\left(\frac{1}{2}\right) \cos 36^{\circ} \\
= & \cos 72^{\circ}-\cos 36^{\circ} \\
= & 2 \sin \left(\frac{72^{\circ}+36^{\circ}}{2}\right) \sin \left(\frac{36^{\circ}-72^{\circ}}{2}\right) \\
= & 2 \sin 54^{\circ} \sin \left(-18^{\circ}\right) \\
= & -2 \sin 54^{\circ} \cdot \sin 18^{\circ} \\
= & -2\left(\frac{\sqrt{5}+1}{4}\right)\left(\frac{\sqrt{5}-1}{4}\right) \\
= & -\frac{1}{8}(5-1)\\
= & -\frac{1}{2}=\text { R.H.S. }
\end{aligned}$

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