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.