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$1+\cos 10^{\circ}+\cos 20^{\circ}+\cos 30^{\circ}=$
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2431 Upvotes
Verified Answer
The correct answer is:
$4 \cos 5^{\circ} \cos 10^{\circ} \cos 15^{\circ}$
We have,
$$
\begin{aligned}
& 1+\cos 10^{\circ}+\cos 20^{\circ}+\cos 30^{\circ} \\
& \quad=\left(1+\cos 10^{\circ}\right)+\left(\cos 20^{\circ}+\cos 30^{\circ}\right) \\
& =2 \cos 5^{\circ}+2 \cos 25^{\circ} \cos 5^{\circ} \\
& =2 \cos 5^{\circ}\left(\cos 5^{\circ}+\cos 25^{\circ}\right) \\
& =2 \cos 25^{\circ}\left(2 \cos 15^{\circ} \cos 10^{\circ}\right) \\
& =4 \cos 5^{\circ} \cos 10^{\circ} \cos 15^{\circ}
\end{aligned}
$$
$$
\begin{aligned}
& 1+\cos 10^{\circ}+\cos 20^{\circ}+\cos 30^{\circ} \\
& \quad=\left(1+\cos 10^{\circ}\right)+\left(\cos 20^{\circ}+\cos 30^{\circ}\right) \\
& =2 \cos 5^{\circ}+2 \cos 25^{\circ} \cos 5^{\circ} \\
& =2 \cos 5^{\circ}\left(\cos 5^{\circ}+\cos 25^{\circ}\right) \\
& =2 \cos 25^{\circ}\left(2 \cos 15^{\circ} \cos 10^{\circ}\right) \\
& =4 \cos 5^{\circ} \cos 10^{\circ} \cos 15^{\circ}
\end{aligned}
$$
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