Understanding the Expression
The expression given is:
\[ \frac{\sin 2A}{1 - \cos 2A} \times \frac{\cos A}{1 - \cos A} \]
This can be simplified using trigonometric identities.
Key Components of the Expression
- Sin and Cosine Functions:
- The sine function represents the ratio of the opposite side to the hypotenuse in a right triangle.
- The cosine function represents the ratio of the adjacent side to the hypotenuse.
- Double Angle Identities:
- Sin 2A = 2 sin A cos A
- Cos 2A = 1 - 2 sin² A
Simplifying the Expression
1. Substituting Sin 2A:
- Replace sin 2A with 2 sin A cos A.
2. Substituting Cos 2A:
- Replace cos 2A with 1 - 2 sin² A in the denominator.
3. Rewriting the Terms:
- The expression becomes:
\[ \frac{2 \sin A \cos A}{1 - (1 - 2 \sin² A)} \times \frac{\cos A}{1 - \cos A} \]
- This simplifies to:
\[ \frac{2 \sin A \cos A}{2 \sin² A} \times \frac{\cos A}{1 - \cos A} \]
4. Further Simplification:
- Cancel out the common factors. The expression reduces to:
\[ \frac{\cos A}{1 - \cos A} \]
Conclusion
The original expression simplifies neatly to a more manageable form. Understanding these identities can greatly aid in solving trigonometric problems efficiently. By breaking down the components and using algebraic manipulation, complex expressions can often be simplified significantly.