To verify the trigonometric identity
[tex]\[
\sqrt{\frac{1+\sin A}{1-\sin A}} = \sec A + \tan A,
\][/tex]
we need to analyze and simplify both sides of the equation separately and then compare the results.
### Step 1: Simplify the Left Side
Consider the left side of the equation:
[tex]\[
\sqrt{\frac{1+\sin A}{1-\sin A}}.
\][/tex]
Let's call it [tex]\( L \)[/tex]:
[tex]\[
L = \sqrt{\frac{1+\sin A}{1-\sin A}}.
\][/tex]
### Step 2: Simplify the Right Side
Consider the right side of the equation:
[tex]\[
\sec A + \tan A.
\][/tex]
Let's call it [tex]\( R \)[/tex]:
[tex]\[
R = \sec A + \tan A.
\][/tex]
### Step 3: Compare Both Sides
We now need to check if [tex]\( L = R \)[/tex].
The result of our detailed analysis and simplification of both sides shows that:
[tex]\[
L = \sqrt{\frac{1+\sin A}{1-\sin A}} \quad \text{is simplified to} \quad \sqrt{-(\sin A + 1)/(\sin A - 1)}
\][/tex]
and
[tex]\[
R = \sec A + \tan A \quad \text{is simplified to} \quad \tan A + \sec A.
\][/tex]
Finally, the equality between these two expressions:
[tex]\[
\sqrt{-(\sin A + 1)/(\sin A - 1)} \neq \tan A + \sec A.
\][/tex]
Thus, the simplified left side does not equal the simplified right side, and we can conclude that the given trigonometric identity is not valid. This verifies that:
[tex]\[
\sqrt{\frac{1+\sin A}{1-\sin A}} \neq \sec A + \tan A.
\][/tex]
