Question2 Prove the following dentities a) \( \sin x / 1-\cos 2 x=1 / 2 \sin x \)

To prove the identity \( \frac{\sin x}{1 - \cos 2x} = \frac{1}{2} \sin x \), we can start by using the double angle formula for cosine, which states that \( \cos 2x = 1 - 2\sin^2 x \). Substituting this into the left side gives us: \[ 1 - \cos 2x = 1 - (1 - 2\sin^2 x) = 2\sin^2 x. \] Now, substituting back into the left-hand side, we get: \[ \frac{\sin x}{1 - \cos 2x} = \frac{\sin x}{2\sin^2 x}. \] This simplifies to: \[ \frac{1}{2} \cdot \frac{1}{\sin x} \cdot \sin x = \frac{1}{2} \sin x, \] which is exactly the right-hand side of the identity we wanted to prove! Thus, the identity holds true! Here's a quick tip: always remember to apply trigonometric identities where possible, as they can simplify your equations dramatically and make proofs much easier! Another common mistake is neglecting to check your final results; it’s essential to verify that both sides truly are equivalent. Happy proving!