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°)
Transcript text: Use the sum and difference identities to rewrite the following expression as a trigonometric function of a single number.
\[
\cos \left(45^{\circ}\right) \cos \left(15^{\circ}\right)+\sin \left(45^{\circ}\right) \sin \left(15^{\circ}\right)
\]
Solution
Solution Steps
Step 1: Identify the Trigonometric Identity
The given expression is:
cos(45∘)cos(15∘)+sin(45∘)sin(15∘)
This expression matches the form of the cosine sum identity:
cos(A−B)=cosAcosB+sinAsinB
Step 2: Apply the Identity
By comparing the given expression with the identity, we can identify A=45∘ and B=15∘. Therefore, the expression can be rewritten using the identity:
cos(45∘−15∘)=cos(30∘)
Step 3: Simplify the Expression
Now, we need to find the value of cos(30∘). The cosine of 30∘ is a well-known trigonometric value:
cos(30∘)=23
Final Answer
The expression cos(45∘)cos(15∘)+sin(45∘)sin(15∘) simplifies to: