Questions: Use the sum and difference identities to rewrite the following expression as a trigonometric function of a single number. cos(45°)cos(15°)+sin(45°)sin(15°)

Use the sum and difference identities to rewrite the following expression as a trigonometric function of a single number.
cos(45°)cos(15°)+sin(45°)sin(15°)
Transcript text: Use the sum and difference identities to rewrite the following expression as a trigonometric function of a single number. \[ \cos \left(45^{\circ}\right) \cos \left(15^{\circ}\right)+\sin \left(45^{\circ}\right) \sin \left(15^{\circ}\right) \]
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Solution

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Solution Steps

Step 1: Identify the Trigonometric Identity

The given expression is:

cos(45)cos(15)+sin(45)sin(15) \cos \left(45^{\circ}\right) \cos \left(15^{\circ}\right) + \sin \left(45^{\circ}\right) \sin \left(15^{\circ}\right)

This expression matches the form of the cosine sum identity:

cos(AB)=cosAcosB+sinAsinB \cos(A - B) = \cos A \cos B + \sin A \sin B

Step 2: Apply the Identity

By comparing the given expression with the identity, we can identify A=45A = 45^\circ and B=15B = 15^\circ. Therefore, the expression can be rewritten using the identity:

cos(4515)=cos(30) \cos(45^\circ - 15^\circ) = \cos(30^\circ)

Step 3: Simplify the Expression

Now, we need to find the value of cos(30)\cos(30^\circ). The cosine of 3030^\circ is a well-known trigonometric value:

cos(30)=32 \cos(30^\circ) = \frac{\sqrt{3}}{2}

Final Answer

The expression cos(45)cos(15)+sin(45)sin(15)\cos \left(45^{\circ}\right) \cos \left(15^{\circ}\right) + \sin \left(45^{\circ}\right) \sin \left(15^{\circ}\right) simplifies to:

32 \boxed{\frac{\sqrt{3}}{2}}

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