Multiple Choice
Expand the expression using the sum & difference identities and simplify.
12. Trigonometric Identities
Sum and Difference Identities
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Find the exact value of the expression.
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Verified step by step guidance1
First, recognize that the expression involves the cosine of an angle, specifically \( \cos \left( \frac{5\pi}{12} \right) \). This angle is not one of the standard angles for which cosine values are typically memorized, so we need to use angle sum or difference identities to find its exact value.
Use the angle sum identity for cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). We can express \( \frac{5\pi}{12} \) as a sum of angles whose cosine and sine values are known, such as \( \frac{5\pi}{12} = \frac{\pi}{4} + \frac{\pi}{6} \).
Calculate \( \cos \left( \frac{\pi}{4} \right) \) and \( \cos \left( \frac{\pi}{6} \right) \). These are standard angles: \( \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \) and \( \cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2} \).
Calculate \( \sin \left( \frac{\pi}{4} \right) \) and \( \sin \left( \frac{\pi}{6} \right) \). These are also standard angles: \( \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \) and \( \sin \left( \frac{\pi}{6} \right) = \frac{1}{2} \).
Substitute these values into the angle sum identity: \( \cos \left( \frac{5\pi}{12} \right) = \cos \left( \frac{\pi}{4} \right) \cos \left( \frac{\pi}{6} \right) - \sin \left( \frac{\pi}{4} \right) \sin \left( \frac{\pi}{6} \right) \). Simplify the expression to find the exact value.
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