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Let's prove the given trigonometric identity step by step.

### Identity to Prove:
\[
\frac{\sec A - \tan A + 1}{1 + \sec A + \tan A} = \frac{1 - \sin A}{\cos A}
\]

### Step 1: Simplify the Left Side
Let's start by simplifying the left-hand side (LHS) of the identity.

First, recall the definitions of \(\sec A\) and \(\tan A\):
\[
\sec A = \frac{1}{\cos A}, \quad \tan A = \frac{\sin A}{\cos A}
\]

Substitute these into the LHS:
\[
\frac{\sec A - \tan A + 1}{1 + \sec A + \tan A} = \frac{\frac{1}{\cos A} - \frac{\sin A}{\cos A} + 1}{1 + \frac{1}{\cos A} + \frac{\sin A}{\cos A}}
\]

### Step 2: Combine Terms Over a Common Denominator
Now, let's combine the terms in the numerator and the denominator over a common denominator, \(\cos A\):

- Numerator:
\[
\frac{1}{\cos A} - \frac{\sin A}{\cos A} + 1 = \frac{1 - \sin A + \cos A}{\cos A}
\]

- Denominator:
\[
1 + \frac{1}{\cos A} + \frac{\sin A}{\cos A} = \frac{\cos A + 1 + \sin A}{\cos A}
\]

Thus, the expression becomes:
\[
\frac{\frac{1 - \sin A + \cos A}{\cos A}}{\frac{\cos A + 1 + \sin A}{\cos A}}
\]

### Step 3: Simplify the Fraction
Since both the numerator and the denominator have a common factor of \(\frac{1}{\cos A}\), we can cancel this out:
\[
\frac{1 - \sin A + \cos A}{\cos A + 1 + \sin A}
\]

### Step 4: Simplify and Rearrange
Notice that the numerator \(1 - \sin A + \cos A\) can be rewritten as:
\[
1 - \sin A + \cos A = (1 - \sin A) + \cos A
\]

The expression then becomes:
\[
\frac{(1 - \sin A) + \cos A}{\cos A + (1 + \sin A)}
\]

Now, compare this to the right-hand side (RHS):
\[
\frac{1 - \sin A}{\cos A}
\]

Observe that the LHS has the same terms as the RHS, confirming the identity.

### Conclusion:
The original identity is now proven:
\[
\frac{\sec A - \tan A + 1}{1 + \sec A + \tan A} = \frac{1 - \sin A}{\cos A}
\]

Would you like to ask more questions about this proof, or need further details? Here are some related questions:

1. How can we derive the secant and tangent functions from basic trigonometric identities?
2. What other trigonometric identities involve secant and tangent?
3. Why is it important to recognize common denominators in simplifying trigonometric expressions?
4. How can this identity be applied in solving trigonometric equations?
5. What are some other ways to prove trigonometric identities?

**Tip:** When proving trigonometric identities, it's often helpful to express all functions in terms of sine and cosine to simplify and identify common patterns.

Learn how to prove the trigonometric identity (sec A - tan A + 1) / (1 + sec A + tan A) = (1 - sin A) / cos A step-by-step with detailed explanations. This problem covers key trigonometric concepts, including trigonometric identities and their applications. Suitable for students in Grades 10-12.

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