Simplify the following trigonometric expression so that the final answer contains only one trigonometric function, and contains no fractions (sin20) (cot20) (sec² 0)

We are given the expression:

\[
(\sin 20^\circ) \cdot (\cot 20^\circ) \cdot (\sec^2 0^\circ)
\]

We need to simplify it such that the final answer contains only one trigonometric function and no fractions. Let's proceed step by step.

### Step 1: Rewrite \(\cot 20^\circ\)

Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). Applying this, we get:

\[
\cot 20^\circ = \frac{\cos 20^\circ}{\sin 20^\circ}
\]

Substituting this into the original expression:

\[
(\sin 20^\circ) \cdot \frac{\cos 20^\circ}{\sin 20^\circ} \cdot \sec^2 0^\circ
\]

### Step 2: Simplify the expression

Now, notice that \(\sin 20^\circ\) in the numerator and denominator cancel each other:

\[
\cos 20^\circ \cdot \sec^2 0^\circ
\]

### Step 3: Simplify \(\sec^2 0^\circ\)

Recall that \(\sec 0^\circ = \frac{1}{\cos 0^\circ}\), and since \(\cos 0^\circ = 1\), we have:

\[
\sec^2 0^\circ = 1^2 = 1
\]

Substituting this into the expression:

\[
\cos 20^\circ \cdot 1 = \cos 20^\circ
\]

### Final Answer:

The simplified expression is:

\[
\cos 20^\circ
\]

Thus, the given trigonometric expression simplifies to \(\cos 20^\circ\).

Would you like further details or have any questions about the steps? Here are some related questions:

1. How do you derive the identity for \(\cot \theta\)?
2. What is the geometric meaning of \(\sec \theta\)?
3. How does \(\sec^2 0^\circ\) simplify to 1?
4. Can you explain how the cancellation of \(\sin 20^\circ\) works?
5. What happens if you replace 20° with another angle, say 45°?

**Tip:** Always check for fundamental identities when simplifying trigonometric expressions; they often help reduce complexity quickly.

Simplify the following trigonometric expression so that the final answer contains only one trigonometric function, and contains no fractions: (sin20) (cot20) (sec² 0)

This tutorial walks through simplifying the trigonometric expression (sin20) (cot20) (sec² 0) to a single function without fractions. Key trigonometric identities and concepts are explained step-by-step, suitable for students in Grades 10-12.

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