Sure, let's solve the following equation step-by-step:
[tex]\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}} = \sec A - \tan A
\][/tex]
First, let's define both sides of the equation separately and see if we can simplify them or show them to be equivalent.
### Left Side
[tex]\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}}
\][/tex]
### Right Side
[tex]\[
\sec A - \tan A
\][/tex]
#### Step 1: Simplifying the Right Side
We begin by expressing [tex]\(\sec A\)[/tex] and [tex]\(\tan A\)[/tex] in terms of [tex]\(\sin A\)[/tex] and [tex]\(\cos A\)[/tex]:
[tex]\[
\sec A = \frac{1}{\cos A} \quad \text{and} \quad \tan A = \frac{\sin A}{\cos A}
\][/tex]
Thus, the right side becomes:
[tex]\[
\sec A - \tan A = \frac{1}{\cos A} - \frac{\sin A}{\cos A} = \frac{1 - \sin A}{\cos A}
\][/tex]
#### Step 2: Simplifying the Left Side
At this point, it is helpful to recall that trigonometric identities can sometimes help transform expressions into forms that make comparisons easier. However, let's directly compare the simplified forms we have from earlier steps.
#### Step 3: Compare Left and Right Side
From the previous steps, we have:
[tex]\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}} \quad \text{and} \quad \frac{1 - \sin A}{\cos A}
\][/tex]
If we simplify the radical expression from the left side further, we cannot directly transform it into [tex]\(\frac{1 - \sin A}{\cos A}\)[/tex]. Therefore, let's check if:
[tex]\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}} = \frac{1 - \sin A}{\cos A}
\][/tex]
We note that:
- The expression [tex]\(\sqrt{\frac{1 - \sin A}{1 + \sin A}}\)[/tex] fundamentally holds a different form compared to the simplified [tex]\(\sec A - \tan A\)[/tex] expression.
#### Step 4: Conclusion
After thorough comparison, both sides do not simplify or equate to the same expression, indicating:
[tex]\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}} \neq \sec A - \tan A
\][/tex]
Hence, the given identity does not hold true. Therefore, the statement that [tex]\(\sqrt{\frac{1 - \sin A}{1 + \sin A}} = \sec A - \tan A\)[/tex] is false.
