Full solution

\frac{1-\sin 2 A}{\cos 2 A}=\frac{1-\tan A}{1+\tan A}

Apply Trigonometric Identities: Use the identity $\sin 2A = 2 \sin A \cos A$ and $\cos 2A = \cos^2 A - \sin^2 A$ . $\frac{1 - \sin 2A}{\cos 2A} = \frac{1 - 2 \sin A \cos A}{\cos^2 A - \sin^2 A}$ .
Utilize Tangent Identity: Use the identity $\tan A = \frac{\sin A}{\cos A}$ . $\frac{1 - 2 \sin A \cos A}{\cos^2 A - \sin^2 A} = \frac{1 - 2(\frac{\sin A}{\cos A}) \cdot \cos A}{\cos^2 A - (\frac{\sin A}{\cos A})^2 \cdot \cos^2 A}$ .
Simplify the Expression: Simplify the expression. $\newline$ $(1 - 2(\frac{\sin A}{\cos A}) \cdot \cos A) / (\cos^2 A - (\frac{\sin A}{\cos A})^2 \cdot \cos^2 A) = (1 - 2 \tan A) / (1 - \tan^2 A).$
Apply Trigonometric Identity: Use the identity $1 - \tan^2 A = (1 - \tan A)(1 + \tan A)$ . $\frac{1 - 2 \tan A}{1 - \tan^2 A} = \frac{1 - 2 \tan A}{(1 - \tan A)(1 + \tan A)}$ .
Factor Out: Factor out $(1 - \tan A)$ from the numerator. $\newline$ $(1 - 2 \tan A) / ((1 - \tan A)(1 + \tan A)) = ((1 - \tan A) - \tan A) / ((1 - \tan A)(1 + \tan A)).$
Cancel Out: Cancel out the $(1 - \tan A)$ term in the numerator and denominator. $\frac{(1 - \tan A) - \tan A}{(1 - \tan A)(1 + \tan A)} = \frac{-\tan A}{1 + \tan A}.$