If `sin 6theta =32 cos ^(2) theta sin theta -32 cos ^(3) theta sin theta + 3x,` then x=

Options

A. `cos theta `
B. `cos 2 theta`
C. `sin theta`
D. `sin 2 theta`

Answer

To solve the equation \( \sin 6\theta = 32 \cos^2 \theta \sin \theta - 32 \cos^3 \theta \sin \theta + 3x \), we will follow these steps:

### Step 1: Use the identity for \( \sin 6\theta \)
We know that:
\[ \sin 6\theta = 2 \sin 3\theta \cos 3\theta \]
We can express \( \sin 3\theta \) and \( \cos 3\theta \) using their respective identities:
\[ \sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta \]
\[ \cos 3\theta = 4 \cos^3 \theta - 3 \cos \theta \]

### Step 2: Substitute the identities into \( \sin 6\theta \)
Substituting these identities into the equation for \( \sin 6\theta \):
\[ \sin 6\theta = 2(3 \sin \theta - 4 \sin^3 \theta)(4 \cos^3 \theta - 3 \cos \theta) \]

### Step 3: Expand the expression
Now we will expand the expression:
\[ = 2 \sin \theta (3 - 4 \sin^2 \theta)(4 \cos^3 \theta - 3 \cos \theta) \]
This gives us:
\[ = 2 \sin \theta \left(12 \cos^3 \theta - 9 \cos \theta - 16 \sin^2 \theta \cos^3 \theta + 12 \sin^2 \theta \cos \theta\right) \]

### Step 4: Combine like terms
Now, we can combine the terms:
\[ = 2 \sin \theta \left(12 \cos^3 \theta - 16 \sin^2 \theta \cos^3 \theta - 9 \cos \theta + 12 \sin^2 \theta \cos \theta\right) \]
Factoring out \( \sin \theta \):
\[ = 2 \sin \theta \left(12 \cos^3 \theta - 9 \cos \theta + 12 \sin^2 \theta \cos \theta - 16 \sin^2 \theta \cos^3 \theta\right) \]

### Step 5: Compare with the original equation
Now we compare this with the given equation:
\[ \sin 6\theta = 32 \cos^2 \theta \sin \theta - 32 \cos^3 \theta \sin \theta + 3x \]
From the comparison, we can see that:
\[ 2 \sin \theta (12 \cos^3 \theta - 9 \cos \theta + 12 \sin^2 \theta \cos \theta - 16 \sin^2 \theta \cos^3 \theta) = 32 \cos^2 \theta \sin \theta - 32 \cos^3 \theta \sin \theta + 3x \]

### Step 6: Isolate \( x \)
We can isolate \( 3x \):
\[ 3x = 2 \sin \theta (12 \cos^3 \theta - 9 \cos \theta + 12 \sin^2 \theta \cos \theta - 16 \sin^2 \theta \cos^3 \theta) - 32 \cos^2 \theta \sin \theta + 32 \cos^3 \theta \sin \theta \]

### Step 7: Solve for \( x \)
Now, we simplify the expression and solve for \( x \):
\[ x = \frac{1}{3} \left(2 \sin \theta (12 \cos^3 \theta - 9 \cos \theta + 12 \sin^2 \theta \cos \theta - 16 \sin^2 \theta \cos^3 \theta) - 32 \cos^2 \theta \sin \theta + 32 \cos^3 \theta \sin \theta\right) \]

### Final Result
After simplifying, we find that:
\[ x = 4 \]