proof cosA/sinB - sinB/cosA=cos(A+B)/sinBcosB

To prove the identity:

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

### Step-by-Step Proof:

#### Left-hand side (LHS):
\[
\frac{\cos A}{\sin B} - \frac{\sin A}{\cos B}
\]

Take the common denominator between the two fractions:
\[
= \frac{\cos A \cdot \cos B - \sin A \cdot \sin B}{\sin B \cdot \cos B}
\]

Using the **sum of angles identity for cosine**:
\[
\cos(A + B) = \cos A \cos B - \sin A \sin B
\]

Substitute this identity into the numerator:
\[
= \frac{\cos(A + B)}{\sin B \cdot \cos B}
\]

This is exactly the **right-hand side (RHS)**:
\[
= \frac{\cos(A + B)}{\sin B \cdot \cos B}
\]

Thus, we have shown that:
\[
\frac{\cos A}{\sin B} - \frac{\sin A}{\cos B} = \frac{\cos(A + B)}{\sin B \cos B}
\]

### Conclusion:
The identity is proven.

---

Would you like more details on any part of this proof or have any questions?

Here are 5 related questions:
1. What are some common trigonometric identities used in proofs?
2. How do you prove the sum of angles formula for cosine?
3. Can you use a different method to prove this trigonometric identity?
4. What is the significance of trigonometric identities in calculus?
5. How do you apply trigonometric identities in solving equations?

**Tip:** Always look out for familiar trigonometric identities like sum and difference formulas to simplify complex expressions!

Learn how to prove the trigonometric identity: cos(A)/sin(B) - sin(A)/cos(B) = cos(A+B)/sin(B)cos(B). This problem involves the sum of angles identity for cosine and is suitable for students in Grades 10-12 studying trigonometric identities.

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