We are given the following trigonometric equation to solve:

\[
\frac{1 - \sec(A)}{1 + \sec(A)} = \frac{1 - \cos(A)}{1 + \cos(A)}
\]

### Step 1: Express \(\sec(A)\) in terms of \(\cos(A)\)
Since \(\sec(A) = \frac{1}{\cos(A)}\), we can substitute this into the equation:

\[
\frac{1 - \frac{1}{\cos(A)}}{1 + \frac{1}{\cos(A)}} = \frac{1 - \cos(A)}{1 + \cos(A)}
\]

### Step 2: Simplify the left-hand side
To simplify, multiply the numerator and denominator by \(\cos(A)\):

\[
\frac{\cos(A)(1 - \frac{1}{\cos(A)})}{\cos(A)(1 + \frac{1}{\cos(A)})} = \frac{\cos(A) - 1}{\cos(A) + 1}
\]

Now the equation becomes:

\[
\frac{\cos(A) - 1}{\cos(A) + 1} = \frac{1 - \cos(A)}{1 + \cos(A)}
\]

### Step 3: Analyze the equation
The left-hand side is \(\frac{\cos(A) - 1}{\cos(A) + 1}\), while the right-hand side is \(\frac{1 - \cos(A)}{1 + \cos(A)}\). Notice that:

\[
1 - \cos(A) = -( \cos(A) - 1)
\]

So the right-hand side is the negative of the left-hand side:

\[
\frac{1 - \cos(A)}{1 + \cos(A)} = -\frac{\cos(A) - 1}{\cos(A) + 1}
\]

Thus, the equation becomes:

\[
\frac{\cos(A) - 1}{\cos(A) + 1} = -\frac{\cos(A) - 1}{\cos(A) + 1}
\]

### Step 4: Solve for \(\cos(A)\)
For this equation to hold, the only way for \(\frac{\cos(A) - 1}{\cos(A) + 1} = -\frac{\cos(A) - 1}{\cos(A) + 1}\) is if:

\[
\cos(A) - 1 = 0
\]

This implies:

\[
\cos(A) = 1
\]

### Step 5: Conclusion
The solution to the equation is:

\[
A = 0 + 2n\pi \quad \text{for any integer} \, n
\]

Would you like more details on any step? Here are 5 related questions:

1. What is the definition of \(\sec(A)\) in trigonometry?
2. How can trigonometric identities help simplify complex equations?
3. What are the implications of \(\cos(A) = 1\) in trigonometry?
4. How would the equation change if \(\cos(A) \neq 1\)?
5. How do transformations affect trigonometric identities?

**Tip**: When simplifying trigonometric equations, always look for basic identities like \(\sec(A) = \frac{1}{\cos(A)}\) to help reduce complexity.

Learn how to solve the trigonometric equation (sec(A)-1)/(sec(A)+1) = (1-cos(A))/(1+cos(A)) using identity simplifications. This solution covers key concepts in trigonometry, including secant and cosine identities, and is suitable for students in Grades 10-12.

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