$$ 6 \quad ( 3 - 3 \sin heta ) ( 3 + 3 \sin heta ) $$

Question

$$ 6 \quad ( 3 - 3 \sin 	heta ) ( 3 + 3 \sin 	heta ) $$

UpStudy AI · Accepted Answer

Let's simplify the expression:

$$
6 \cdot (3 - 3 \sin \theta) \cdot (3 + 3 \sin \theta)
$$

**Step 1: Factor Out Common Terms**

First, factor out the common factor of 3 from both binomials:

$$
= 6 \cdot 3(1 - \sin \theta) \cdot 3(1 + \sin \theta)
$$

**Step 2: Multiply the Constants**

Multiply the constants together:

$$
= 6 \cdot 3 \cdot 3 \cdot (1 - \sin \theta)(1 + \sin \theta)
$$
$$
= 54 \cdot (1 - \sin \theta)(1 + \sin \theta)
$$

**Step 3: Apply the Difference of Squares Formula**

Recall that $$(a - b)(a + b) = a^2 - b^2$$. Apply this to the binomials:

$$
= 54 \cdot (1^2 - (\sin \theta)^2)
$$
$$
= 54 \cdot (1 - \sin^2 \theta)
$$

**Step 4: Use a Pythagorean Identity**

We know that $$1 - \sin^2 \theta = \cos^2 \theta$$. Substitute this in:

$$
= 54 \cdot \cos^2 \theta
$$

**Final Simplified Form:**

$$
54 \cos^2 \theta
$$

So, the simplified form of the expression $$6 \cdot (3 - 3 \sin \theta) \cdot (3 + 3 \sin \theta)$$ is:

$$
54 \cos^2 \theta
$$
The simplified form of the expression is $$54 \cos^2 \theta$$.

\( 6 \quad ( 3 - 3 \sin \theta ) ( 3 + 3 \sin \theta ) \)