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°)

Solution Steps

The given expression is:

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

This expression matches the form of the cosine sum identity:

$$\cos(A - B) = \cos A \cos B + \sin A \sin B$$

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

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

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

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

Final Answer