sin theta times tan theta plus sec theta.

To simplify the expression \(\sin(\theta) \cdot \tan(\theta) + \sec(\theta)\), let's break it down step by step:

1. First, recall that:
   - \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
   - \(\sec(\theta) = \frac{1}{\cos(\theta)}\)

2. Substitute these identities into the expression:

\[
\sin(\theta) \cdot \frac{\sin(\theta)}{\cos(\theta)} + \frac{1}{\cos(\theta)}
\]

3. Simplify the terms:

\[
\frac{\sin^2(\theta)}{\cos(\theta)} + \frac{1}{\cos(\theta)}
\]

4. Combine the two terms since they have the same denominator:

\[
\frac{\sin^2(\theta) + 1}{\cos(\theta)}
\]

5. Now use the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\), which gives:

\[
\frac{1}{\cos(\theta)} = \sec(\theta)
\]

Thus, the simplified expression is:

\[
\sec(\theta)
\]

### Do you need more details or have any questions?

Here are 5 related questions:
1. How do you prove trigonometric identities?
2. What is the importance of the Pythagorean identity in trigonometry?
3. How do you simplify trigonometric expressions with multiple terms?
4. What is the reciprocal relationship between secant and cosine?
5. How can you solve trigonometric equations involving secant?

**Tip**: Always remember to use fundamental trigonometric identities like the Pythagorean identity to simplify complex expressions.

Learn how to simplify the trigonometric expression sin(θ) * tan(θ) + sec(θ) using key trigonometric identities such as the tangent and secant definitions, and the Pythagorean identity. This step-by-step guide is ideal for students in Grades 10-12.

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