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Question
The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is
Options
sin 2A
cos 2A
cos 3A
sin 3A
Solution
cos 2A
\[\cos(36^\circ - A)\cos(36^\circ + A) + \cos(54^\circ + A)\cos(54^\circ - A)\]
\[ = \cos(36^\circ - A) \cos(36^\circ + A) + \sin\left[ 90^\circ - (54^\circ + A) \right] \sin\left[ 90^\circ - (54^\circ - A) \right] \left[\text{ Since }\sin(90^\circ - \theta) = \cos\theta \right]\]
\[ = \cos(36^\circ - A)\cos(36^\circ + A) + \sin(36^\circ - A)\sin(36^\circ + A)\]
\[ = \cos(36^\circ + A - 36^\circ + A) \left[\text{ Using }\cos(A - B)\text{ formula }\right]\]
\[ = \cos 2A\]
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