https://file.aiquickdraw.com/mathbot/file/1726817004107ztxzpuwi.jpg

The expression shown in the image is:

\[
\sec \theta - \tan \theta \sin \theta
\]

Let's simplify it step by step.

### Step 1: Recall trigonometric identities.
We know that:
\[
\sec \theta = \frac{1}{\cos \theta}
\]
and
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]

### Step 2: Substitute the identity for \(\tan \theta\).
The given expression becomes:
\[
\sec \theta - \left(\frac{\sin \theta}{\cos \theta}\right)\sin \theta
\]

### Step 3: Simplify the second term.
The second term simplifies to:
\[
\frac{\sin^2 \theta}{\cos \theta}
\]

Thus, the expression now becomes:
\[
\frac{1}{\cos \theta} - \frac{\sin^2 \theta}{\cos \theta}
\]

### Step 4: Combine the terms.
Since both terms have the same denominator, we can combine them:
\[
\frac{1 - \sin^2 \theta}{\cos \theta}
\]

### Step 5: Use the Pythagorean identity.
We use the Pythagorean identity:
\[
1 - \sin^2 \theta = \cos^2 \theta
\]

Thus, the expression simplifies to:
\[
\frac{\cos^2 \theta}{\cos \theta}
\]

### Step 6: Simplify further.
Finally, simplify the fraction:
\[
\cos \theta
\]

So, the simplified form of the expression is:
\[
\boxed{\cos \theta}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this:
1. How do you derive the Pythagorean identity \(1 - \sin^2 \theta = \cos^2 \theta\)?
2. What is the relationship between \(\sec \theta\) and \(\cos \theta\)?
3. Can you apply similar steps to simplify the expression \(\sec^2 \theta - \tan^2 \theta\)?
4. How does the simplification of trigonometric expressions help in solving trigonometric equations?
5. What are other common trigonometric identities used in simplifications?

**Tip**: Always look for common identities like the Pythagorean identities to simplify complex trigonometric expressions.

Simplify the expression secθ - tanθ sinθ

Learn how to simplify the trigonometric expression secθ - tanθ sinθ using basic trigonometric identities. This problem covers key concepts in trigonometry, such as secant and tangent functions, and the Pythagorean identity, suitable for Grades 10-12 students.

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