https://file.aiquickdraw.com/mathbot/file/172553748542739t9pzv5.jpg

The expression from the image is:

\[
\text{Simplify: } \sec(A) + \tan(A) \times (1 - \sin(A))
\]

Let's work through this step by step.

### Step 1: Recall the identities for secant and tangent
- \(\sec(A) = \frac{1}{\cos(A)}\)
- \(\tan(A) = \frac{\sin(A)}{\cos(A)}\)

### Step 2: Expand the expression
The given expression is:
\[
\sec(A) + \tan(A) \times (1 - \sin(A))
\]
Substituting the identities for \(\sec(A)\) and \(\tan(A)\), we get:
\[
\frac{1}{\cos(A)} + \frac{\sin(A)}{\cos(A)} \times (1 - \sin(A))
\]

### Step 3: Simplify the second term
The second term is \(\frac{\sin(A)}{\cos(A)} \times (1 - \sin(A))\), which simplifies to:
\[
\frac{\sin(A)(1 - \sin(A))}{\cos(A)}
\]

### Step 4: Combine the terms
Now the expression becomes:
\[
\frac{1}{\cos(A)} + \frac{\sin(A)(1 - \sin(A))}{\cos(A)}
\]
Since both terms have the same denominator \(\cos(A)\), we can combine them:
\[
\frac{1 + \sin(A)(1 - \sin(A))}{\cos(A)}
\]

### Step 5: Expand and simplify
Now expand \(\sin(A)(1 - \sin(A))\):
\[
\sin(A)(1 - \sin(A)) = \sin(A) - \sin^2(A)
\]
Substitute this back into the expression:
\[
\frac{1 + \sin(A) - \sin^2(A)}{\cos(A)}
\]

### Final Simplified Expression:
Thus, the simplified form of the expression is:
\[
\frac{1 + \sin(A) - \sin^2(A)}{\cos(A)}
\]

Would you like further clarification or details on any of the steps?

### Related Questions:
1. How do you derive the secant and tangent identities from basic trigonometry?
2. What are some common applications of trigonometric identities in calculus?
3. How can you simplify trigonometric expressions involving multiple terms?
4. Why is it important to have the same denominator when combining terms in trigonometric expressions?
5. What other trigonometric identities could be used to simplify similar expressions?

### Tip:
When simplifying trigonometric expressions, always check if terms can be rewritten using basic identities like \(\sin^2(A) + \cos^2(A) = 1\). This can often lead to further simplifications.

Simplify the expression: sec(A) + tan(A) * (1 - sin(A))

Learn how to simplify the trigonometric expression sec(A) + tan(A) * (1 - sin(A)) using basic trigonometric identities. Follow a detailed step-by-step solution for high school students in Grades 9-12.

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