Without the use of tables or a calculator prove that \( \frac{1-\cos 2 A}{\sin 2 A}=\tan A \)

To prove that \( \frac{1 - \cos 2A}{\sin 2A} = \tan A \) without the use of tables or a calculator, we can start with some basic trigonometric identities. We know that: - The double angle formulas are: \[ \cos 2A = 1 - 2\sin^2 A \] \[ \sin 2A = 2\sin A \cos A \] Now, substituting the expression for \( \cos 2A \) into our original equation gives: \[ 1 - \cos 2A = 1 - (1 - 2\sin^2 A) = 2\sin^2 A \] Next, substitute this and the expression for \( \sin 2A \) into the left side of our equation: \[ \frac{1 - \cos 2A}{\sin 2A} = \frac{2\sin^2 A}{2\sin A \cos A} \] We can simplify this: \[ \frac{2\sin^2 A}{2\sin A \cos A} = \frac{\sin A}{\cos A} = \tan A \] Thus, we have shown that: \[ \frac{1 - \cos 2A}{\sin 2A} = \tan A \] This completes the proof!