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

<p><br />To prove the identity \(\cot \theta = \frac{\sin 2 \theta}{1 - \cos 2 \theta}\), we will use trigonometric identities.<br /><br />1. **Double Angle Formulas**: We need the double angle formulas for sine and cosine. They are:<br /> - \(\sin 2\theta = 2\sin \theta \cos \theta\)<br /> - \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)<br /> These can be derived from the sum formulas for sine and cosine.<br /><br />2. **Cotangent Definition**: Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).<br /><br />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).<br /> - Begin with \(\frac{\sin 2 \theta}{1 - \cos 2 \theta}\).<br /> - Substitute the double angle formulas: \(\frac{2\sin \theta \cos \theta}{1 - (\cos^2 \theta - \sin^2 \theta)}\).<br /> - Simplify the denominator: \(1 - \cos^2 \theta + \sin^2 \theta = 1 - \cos^2 \theta + (1 - \cos^2 \theta) = 2(1 - \cos^2 \theta)\).<br /> - So, \(\frac{2\sin \theta \cos \theta}{2(1 - \cos^2 \theta)}\).<br /> - Simplify further: \(\frac{\sin \theta \cos \theta}{1 - \cos^2 \theta}\).<br /><br />4. **Use the Pythagorean Identity**: The Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) can be rearranged to \(\sin^2 \theta = 1 - \cos^2 \theta\).<br /> - Substitute \(\sin^2 \theta\) for \(1 - \cos^2 \theta\) in the denominator: \(\frac{\sin \theta \cos \theta}{\sin^2 \theta}\).<br /><br />5. **Final Simplification**: The expression now becomes \(\frac{\cos \theta}{\sin \theta}\), which is the definition of \(\cot \theta\).<br /><br />Therefore, we have shown that \(\cot \theta = \frac{\sin 2 \theta}{1 - \cos 2 \theta}\), proving the identity.</p>