Sure, let's verify the equation step-by-step. We're given:
[tex]\[
\frac{\sec A -\tan A }{\sec A +\tan A } = 1 - 2 \sec A \cdot \tan A + 2 \tan^2 A
\][/tex]
Firstly, let's take the left-hand side (LHS) of the equation:
[tex]\[
\frac{\sec A - \tan A}{\sec A + \tan A}
\][/tex]
On simplifying the LHS, we achieved the following expression:
[tex]\[
LHS = \frac{1 - \sin A}{\sin A + 1}
\][/tex]
Next, let's take the right-hand side (RHS) of the equation:
[tex]\[
1 - 2 \sec A \cdot \tan A + 2 \tan^2 A
\][/tex]
On simplifying the RHS, we achieved the following expression:
[tex]\[
RHS = 2 \tan^2 A - 2 \tan A \cdot \sec A + 1
\][/tex]
Now, let's compare the simplified forms of LHS and RHS to check if they are equal:
[tex]\[
LHS = \frac{1 - \sin A}{\sin A + 1}
\][/tex]
[tex]\[
RHS = 2 \tan^2 A - 2 \tan A \cdot \sec A + 1
\][/tex]
From the results, we can observe that:
[tex]\[
\frac{1 - \sin A}{\sin A + 1} \neq 2 \tan^2 A - 2 \tan A \cdot \sec A + 1
\][/tex]
Thus, the given equation doesn't hold true as the simplified left-hand side does not equal the simplified right-hand side.
