Use a calculator to evaluate each of the following (a) \( \cos (A-B) \) (b) \( \cos A-\cos B \) (c) \( \cos A \cdot \cos B+\sin A \cdot \sin B \)

It looks like you're looking to evaluate the following trigonometric expressions: (a) \( \cos (A - B) \) (b) \( \cos A - \cos B \) (c) \( \cos A \cdot \cos B + \sin A \cdot \sin B \) However, to perform numerical evaluations using a calculator, specific values for angles \( A \) and \( B \) are required. Without these values, we can instead simplify and express these expressions using trigonometric identities. Here's how each part can be approached: --- ### (a) \( \cos (A - B) \) **Using the Cosine Difference Identity:** The cosine of a difference of two angles can be expanded using the following identity: \[ \cos (A - B) = \cos A \cos B + \sin A \sin B \] **Simplified Form:** \[ \cos (A - B) = \cos A \cos B + \sin A \sin B \] --- ### (b) \( \cos A - \cos B \) **Using the Sum-to-Product Identity:** The difference of cosines can be expressed as a product using the sum-to-product identities: \[ \cos A - \cos B = -2 \sin\left( \frac{A + B}{2} \right) \sin\left( \frac{A - B}{2} \right) \] **Simplified Form:** \[ \cos A - \cos B = -2 \sin\left( \frac{A + B}{2} \right) \sin\left( \frac{A - B}{2} \right) \] --- ### (c) \( \cos A \cdot \cos B + \sin A \cdot \sin B \) **Using the Cosine Difference Identity:** This expression is identical to the right-hand side of the cosine difference identity mentioned in part (a): \[ \cos A \cos B + \sin A \sin B = \cos (A - B) \] **Simplified Form:** \[ \cos A \cos B + \sin A \sin B = \cos (A - B) \] --- ### Summary of Relationships - **Part (a)** and **Part (c)** are essentially the same due to the cosine difference identity: \[ \cos (A - B) = \cos A \cos B + \sin A \sin B \] - **Part (b)** expresses the difference of two cosines as a product of sines, using the sum-to-product identity: \[ \cos A - \cos B = -2 \sin\left( \frac{A + B}{2} \right) \sin\left( \frac{A - B}{2} \right) \] If you have specific values for \( A \) and \( B \), you can plug them into these simplified forms and use a calculator to obtain numerical results.