What is the value of `sin8theta+sin6theta` ?

Options

A. `2sin7thetacos7theta`
B. `2sin7thetacostheta`
C. `2sinthetacos7theta`
D. `2sin4thetacos3theta`

Answer

To find the value of \( \sin 8\theta + \sin 6\theta \), we can use the sine addition formula. Here are the steps to solve the problem:

### Step 1: Use the Sine Addition Formula
The formula for the sum of two sine functions is:
\[ \sin c + \sin d = 2 \sin\left(\frac{c + d}{2}\right) \cos\left(\frac{c - d}{2}\right) \]
In our case, let \( c = 8\theta \) and \( d = 6\theta \).

### Step 2: Calculate \( c + d \) and \( c - d \)
Now, we need to calculate \( c + d \) and \( c - d \):
\[ c + d = 8\theta + 6\theta = 14\theta \]
\[ c - d = 8\theta - 6\theta = 2\theta \]

### Step 3: Substitute into the Formula
Now we substitute \( c + d \) and \( c - d \) back into the sine addition formula:
\[ \sin 8\theta + \sin 6\theta = 2 \sin\left(\frac{14\theta}{2}\right) \cos\left(\frac{2\theta}{2}\right) \]

### Step 4: Simplify the Expression
Now simplify the fractions:
\[ \sin 8\theta + \sin 6\theta = 2 \sin(7\theta) \cos(\theta) \]

### Final Answer
Thus, the value of \( \sin 8\theta + \sin 6\theta \) is:
\[ \sin 8\theta + \sin 6\theta = 2 \sin(7\theta) \cos(\theta) \]

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