First slide

Trigonometric transformations

Question

The value of $1 + \cos 56^{\circ} + \cos 58^{\circ} - \cos 66^{\circ}$ is equal to

Moderate

A

$2 \cos 28^{\circ} \cos 29^{\circ} \cos 33^{\circ}$
B

$4 \cos 28^{\circ} \cos 29^{\circ} \sin 33^{\circ}$
C

$4 \cos 28^{\circ} \cos 29^{\circ} \cos 33^{\circ}$
D

$2 \cos 28^{\circ} \cos 29^{\circ} \sin 33^{\circ}$

Solution

$\begin{aligned} 1 + \cos 56^{\circ} + \cos 58^{\circ} - \cos 66^{\circ} \\ = 2 \cos^{2} 28^{\circ} + 2 \sin 62^{\circ} \sin 4^{\circ} \\ = 2 \cos^{2} 28^{\circ} + 2 \cos 28^{\circ} \cos 86^{\circ} \\ = 2 \cos 28^{\circ} (\cos 28^{\circ} + \cos 86^{\circ}) \\ = 2 \cos 28^{\circ} (2 \cos 57^{\circ} \cos 29^{\circ}) \\ = 4 \cos 28^{\circ} \cos 29^{\circ} \cos 33^{\circ} \end{aligned}$

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