how do i prove the formula for cos(A-B) = cosACosB+SinASinB

To prove the trigonometric identity \( \cos(A - B) = \cos A \cos B + \sin A \sin B \), we can use the angle addition formula for cosine. The angle addition formula states that:

\[
\cos(X + Y) = \cos X \cos Y - \sin X \sin Y
\]

To derive the identity for \( \cos(A - B) \), we can substitute \( B \) with \( -B \) in the angle addition formula:

\[
\cos(A - B) = \cos A \cos(-B) - \sin A \sin(-B)
\]

Now, we use the following trigonometric identities for the negative angles:

\[
\cos(-B) = \cos B \quad \text{and} \quad \sin(-B) = -\sin B
\]

Substituting these into the formula:

\[
\cos(A - B) = \cos A \cos B - \sin A (-\sin B)
\]

Simplifying this, we get:

\[
\cos(A - B) = \cos A \cos B + \sin A \sin B
\]

Thus, we've proven the identity:

\[
\cos(A - B) = \cos A \cos B + \sin A \sin B
\]

Would you like more details or have any questions about this proof?

Here are five related questions to deepen your understanding:

1. How can the angle addition formulas for sine and cosine be used to derive other trigonometric identities?
2. What is the formula for \( \cos(A + B) \) and how does it relate to the formula for \( \cos(A - B) \)?
3. How can the identity \( \cos(A - B) = \cos A \cos B + \sin A \sin B \) be applied in solving trigonometric equations?
4. Can you prove the sine subtraction formula \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)?
5. How does the identity \( \cos(A - B) = \cos A \cos B + \sin A \sin B \) relate to the geometric interpretation of cosine?

**Tip:** When proving trigonometric identities, always keep fundamental identities (like Pythagorean identities) in mind, as they often simplify the process.

Learn how to prove the trigonometric identity cos(A - B) = cos A cos B + sin A sin B with a detailed step-by-step solution. This problem covers key trigonometric concepts, including the angle addition formula for cosine. Suitable for high school students and anyone interested in trigonometry.

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