Prove the following identity. cot theta=(sin 2theta)/(1-cos 2theta)

To prove the identity , we will use trigonometric identities.1. **Double Angle Formulas**: We need the double angle formulas for sine and cosine. They are: - - These can be derived from the sum formulas for sine and cosine.2. **Cotangent Definition**: Recall that .3. **Start with the Right-Hand Side (RHS)**: We start with the RHS of the identity and try to manipulate it to reach the left-hand side (LHS). - Begin with . - Substitute the double angle formulas: \(\frac{2\sin \theta \cos \theta}{1 - (\cos^2 \theta - \sin^2 \theta)}\). - Simplify the denominator: \(1 - \cos^2 \theta + \sin^2 \theta = 1 - \cos^2 \theta + (1 - \cos^2 \theta) = 2(1 - \cos^2 \theta)\). - So, \(\frac{2\sin \theta \cos \theta}{2(1 - \cos^2 \theta)}\). - Simplify further: .4. **Use the Pythagorean Identity**: The Pythagorean identity can be rearranged to . - Substitute for in the denominator: .5. **Final Simplification**: The expression now becomes , which is the definition of .Therefore, we have shown that , proving the identity.