3Cosθ + Cos3θ————————3Sinθ - Sin3θ

You have the following expression:

\[
\frac{3 \cos \theta + \cos 3\theta}{3 \sin \theta - \sin 3\theta}
\]

We can simplify this by using the trigonometric identities for \( \cos 3\theta \) and \( \sin 3\theta \).

### Trigonometric Identities:
1. \( \cos 3\theta = 4\cos^3\theta - 3\cos\theta \)
2. \( \sin 3\theta = 3\sin\theta - 4\sin^3\theta \)

Now, substitute these into the given expression:

\[
\frac{3 \cos \theta + (4\cos^3\theta - 3\cos\theta)}{3 \sin \theta - (3\sin\theta - 4\sin^3\theta)}
\]

### Simplifying:
1. For the numerator:  
\[
3 \cos \theta + 4\cos^3\theta - 3\cos\theta = 4\cos^3\theta
\]
So, the numerator becomes \( 4\cos^3\theta \).

2. For the denominator:  
\[
3 \sin \theta - (3 \sin \theta - 4 \sin^3 \theta) = 4 \sin^3 \theta
\]
So, the denominator becomes \( 4 \sin^3 \theta \).

### Final expression:
\[
\frac{4 \cos^3 \theta}{4 \sin^3 \theta} = \left( \frac{\cos \theta}{\sin \theta} \right)^3 = \cot^3 \theta
\]

So, the simplified form of the expression is:
\[
\cot^3 \theta
\]

Would you like further details or have any questions?

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Here are 5 related questions:
1. How do we derive the identity for \( \cos 3\theta \) and \( \sin 3\theta \)?
2. What are some common uses of the cotangent function in trigonometry?
3. How would you simplify expressions involving higher powers of trigonometric functions?
4. Can you simplify an expression involving \( \tan 3\theta \)?
5. How does the simplification of this expression change if \( \theta = 45^\circ \)?

**Tip:** Memorizing standard trigonometric identities can speed up solving complex expressions.

Simplify the trigonometric expression 3Cosθ + Cos3θ / 3Sinθ - Sin3θ using standard identities. This step-by-step solution applies formulas for Cos(3θ) and Sin(3θ), reducing the expression to Cot^3(θ). Perfect for students in Grades 10-12 learning trigonometric simplification techniques.

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