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Question
Write the value of \[\cos^2 76° + \cos^2 16° - \cos 76° \cos 16°\]
Solution
\[We have, \]
\[ \cos^2 76° + \cos^2 16° - \cos76° \cos16° \]
\[ = \frac{1}{2}\left[ 1 + \cos2\left( 76 \right)° + 1 + \cos2\left( 16 \right)° - \cos\left( 76 + 16 \right)° - \cos\left( 76 - 16 \right)° \right]\]
` [ ∵ 2 cos ^2 theta = 1 + cos 2 theta and 2 cos A cos B = cos ( A +B ) + cos ( A-B)]`
\[ = \frac{1}{2}\left[ 2 + \cos152° + \cos32° - \cos92° - \frac{1}{2} \right]\]
\[ = \frac{1}{2}\left[ \frac{3}{2} - \cos\left( 180 - 152° \right) + \cos32° - \cos92° \right]\]
\[= \frac{1}{2}\left[ \frac{3}{2} - \cos28° + 2\sin\frac{92° + 32° }{2} \sin\frac{92° - 32° }{2} \right]\]
\[ \left[ \text{ cos } C - \text{ cos } D = 2\sin\frac{C + D}{2}\sin\frac{D - C}{2} \right]\]
\[ = \frac{1}{2}\left[ \frac{3}{2} - \cos28° + 2\sin\frac{124° }{2} \sin\frac{60° }{2} \right]\]
\[ = \frac{1}{2}\left[ \frac{3}{2} - \cos28° + 2\sin62° \sin30° \right]\]
\[ = \frac{1}{2}\left[ \frac{3}{2} - \cos28° + 2\sin62° \times \frac{1}{2} \right]\]
\[ = \frac{1}{2}\left[ \frac{3}{2} - \cos28° + \sin62° \right]\]
\[ = \frac{1}{2}\left[ \frac{3}{2} - \cos28° + \sin\left( 90 - 28 \right)° \right]\]
\[ = \frac{1}{2}\left[ \frac{3}{2} - \cos28° + \cos28° \right]\]
\[ = \frac{3}{4}\]
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