`(sin7theta+6sin5theta+17sin3theta+12sintheta)/(sin6theta+5sin4theta+12sin2theta)` is equal to

Options

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

Answer

To solve the expression \(\frac{\sin 7\theta + 6\sin 5\theta + 17\sin 3\theta + 12\sin \theta}{\sin 6\theta + 5\sin 4\theta + 12\sin 2\theta}\), we can follow these steps:

### Step 1: Define the Numerator and Denominator
Let:
- \(N = \sin 7\theta + 6\sin 5\theta + 17\sin 3\theta + 12\sin \theta\) (Numerator)
- \(D = \sin 6\theta + 5\sin 4\theta + 12\sin 2\theta\) (Denominator)

### Step 2: Rewrite the Numerator
We can rewrite the numerator \(N\) by splitting the coefficients:
\[ N = \sin 7\theta + (5 + 1)\sin 5\theta + (12 + 5)\sin 3\theta + 12\sin \theta \]
This gives us:
\[ N = \sin 7\theta + 5\sin 5\theta + \sin 5\theta + 12\sin 3\theta + 5\sin 3\theta + 12\sin \theta \]

### Step 3: Group Terms
Now, we can group the terms:
\[ N = \sin 7\theta + 5\sin 5\theta + 12\sin 3\theta + \sin 5\theta + 12\sin \theta \]
This can be factored as:
\[ N = \sin 7\theta + 5(\sin 5\theta + 2\sin 3\theta) + 12\sin \theta \]

### Step 4: Apply the Sine Addition Formula
Using the identity \(\sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\), we can simplify:
1. For \(\sin 7\theta + \sin 5\theta\):
\[ = 2\sin\left(\frac{7\theta + 5\theta}{2}\right)\cos\left(\frac{7\theta - 5\theta}{2}\right) = 2\sin(6\theta)\cos(\theta) \]
2. For \(5\sin 5\theta + 12\sin 3\theta\):
\[ = 5\sin 5\theta + 12\sin 3\theta \]
(This part can be left as is for now.)

### Step 5: Combine Terms
Now, we can express the numerator \(N\) as:
\[ N = 2\sin(6\theta)\cos(\theta) + 5\sin 5\theta + 12\sin 3\theta + 12\sin \theta \]

### Step 6: Factor Out Common Terms
We can factor out \(2\cos(\theta)\) from the numerator:
\[ N = 2\cos(\theta)(\sin(6\theta) + 5\sin(4\theta) + 12\sin(2\theta)) \]

### Step 7: Recognize the Denominator
Notice that the denominator \(D\) is:
\[ D = \sin 6\theta + 5\sin 4\theta + 12\sin 2\theta \]
Thus, we can write:
\[ N = 2\cos(\theta)D \]

### Step 8: Simplify the Expression
Now, substituting back into the original expression:
\[ \frac{N}{D} = \frac{2\cos(\theta)D}{D} = 2\cos(\theta) \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{\sin 7\theta + 6\sin 5\theta + 17\sin 3\theta + 12\sin \theta}{\sin 6\theta + 5\sin 4\theta + 12\sin 2\theta} = 2\cos(\theta) \]