Evaluate the following expression in terms of trigonometric ratios. $$\frac{{\sec A - \tan A}}{{\sec A + \tan A}}$$

A. 1 + 2tan²A - 2secAtanA

B. 1 + 2sec²A - 2secAtanA

C. 1 + 2sec²A + 2secAtanA

D. 1 + 2tan²A + 2secAtanA

Answer: Option A

Solution(By Examveda Team)

$$\eqalign{ & \frac{{\sec A - \tan A}}{{\sec A + \tan A}} \cr & = \frac{{\sec A - \tan A}}{{\sec A + \tan A}} \times \frac{{\sec A - \tan A}}{{\sec A - \tan A}} \cr & = \frac{{{{\left( {\sec A - \tan A} \right)}^2}}}{{{{\sec }^2}A - {{\tan }^2}A}} \cr & = \frac{{{{\sec }^2}A + {{\tan }^2}A - 2.\sec A.\tan A}}{1} \cr & = 1 + {\tan ^2}A + {\tan ^2}A - 2.\sec A.\tan A \cr & = 1 + 2{\tan ^2}A - 2.\sec A.\tan A \cr} $$