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.