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Question
The value of \[\cos \left( 36° - A \right) \cos \left( 36° + A \right) + \cos \left( 54° - A \right) \cos \left( 54° + A \right)\] is
Options
cos 2A
sin 2A
cos A
0
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Solution
cos 2A
\[ = \cos\left[ 90° - \left( 54° + A \right) \right]\cos\left[ 90° - \left( 54° - A \right) \right] + \cos \left( 54° - A \right)\cos\left( 54° + A \right)\]
\[ = \sin\left( 54° + A \right)\sin\left( 54° - A \right) + \cos\left( 54° - A \right)\cos\left( 54° + A \right) \left[ \cos\left( 90° - \theta \right) = \sin\theta \right]\]
\[ = \cos\left( 54° + A - 54° + A \right) \left[ \cos\left( A - B \right) = cosAcosB + sinAsinB \right] \]
\[ = \cos 2A\]
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