To determine if the given equation [tex]\(\sqrt{\frac{1-\sin A}{1+\sin A}} = \sec A - \tan A\)[/tex] is valid, we will analyze both sides step-by-step.
### Step 1: Simplify the Left-Hand Side
The left-hand side (LHS) of the equation is:
[tex]\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}}
\][/tex]
### Step 2: Simplify the Right-Hand Side
The right-hand side (RHS) of the equation is:
[tex]\[
\sec A - \tan A
\][/tex]
We know that [tex]\(\sec A = \frac{1}{\cos A}\)[/tex] and [tex]\(\tan A = \frac{\sin A}{\cos A}\)[/tex]. Substituting these into the RHS gives us:
[tex]\[
\sec A - \tan A = \frac{1}{\cos A} - \frac{\sin A}{\cos A} = \frac{1 - \sin A}{\cos A}
\][/tex]
Upon confirming this, we need to determine if these two expressions are indeed equal.
### Step 3: Equate the Simplified Expressions
So we have:
[tex]\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}} \quad \text{and} \quad \frac{1 - \sin A}{\cos A}
\][/tex]
We are trying to see if:
[tex]\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}} = \frac{1 - \sin A}{\cos A}
\][/tex]
### Step 4: Verify the Equality
To affirm this equation, we need to square both sides and cross-multiply:
[tex]\[
\left(\sqrt{\frac{1 - \sin A}{1 + \sin A}}\right)^2 = \left(\frac{1 - \sin A}{\cos A}\right)^2
\][/tex]
This yields:
[tex]\[
\frac{1 - \sin A}{1 + \sin A} = \frac{(1 - \sin A)^2}{\cos^2 A}
\][/tex]
Cross-multiplying both sides gives us:
[tex]\[
(1 - \sin A) \cos^2 A = (1 + \sin A) (1 - \sin A)^2
\][/tex]
Expanding the right-hand side:
[tex]\[
(1 + \sin A) (1 - \sin A)^2 = (1 + \sin A)(1 - 2\sin A + \sin^2 A)
\][/tex]
Simplifying further on the left-hand side and right-hand side:
[tex]\[
(1 - \sin A) \cos^2 A \quad \text{and} \quad (1 + \sin A)(1 - 2\sin A + \sin^2 A)
\][/tex]
By computation, it turns that:
- The left-hand side is [tex]\(\cos^2 A - \sin A \cos^2 A\)[/tex].
- The right-hand side calculation resolves to a different trigonometric identity.
After inspecting the calculations, it turns out that:
[tex]\(\sqrt{\frac{1 - \sin A}{1 + \sin A}}\)[/tex]
is not directly equal to [tex]\(\sec A - \tan A\)[/tex]. Therefore, they simplify differently and are not equal.
### Conclusion:
After verifying, we note the equation:
[tex]\[
\sqrt{\frac{1 - \sin A}{1 + \sin A}} \neq \sec A - \tan A
\][/tex]
Thus, these two expressions are not identical.
