If `a=tan27theta-tantheta` and `b=(sin theta)/(cos3theta)+(sin3theta)/(cos9theta)+(sin9theta)/(cos27theta),` then

Options

A. `a=b`
B. `a=2b`
C. `b=2a`
D. `a+b=2`

Answer

To solve the problem, we need to express \( b \) in terms of \( a \) and find the relationship between them. Let's break down the solution step by step.

### Step 1: Express \( b \) in terms of trigonometric identities
We start with:
\[ b = \frac{\sin \theta}{\cos 3\theta} + \frac{\sin 3\theta}{\cos 9\theta} + \frac{\sin 9\theta}{\cos 27\theta} \]

### Step 2: Rewrite each term using the sine double angle identity
Using the identity \( \sin A = 2 \sin \frac{A}{2} \cos \frac{A}{2} \), we can rewrite the first term:
\[ \frac{\sin \theta}{\cos 3\theta} = \frac{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}}{\cos 3\theta} \]

### Step 3: Simplify using tangent
We can express the first term as:
\[ \frac{\sin \theta}{\cos 3\theta} = \tan \theta \cdot \frac{1}{\cos 3\theta} \]
This allows us to express it in terms of tangent.

### Step 4: Repeat for the other terms
Similarly, we can express the other two terms:
\[ \frac{\sin 3\theta}{\cos 9\theta} = \tan 3\theta \cdot \frac{1}{\cos 9\theta} \]
\[ \frac{\sin 9\theta}{\cos 27\theta} = \tan 9\theta \cdot \frac{1}{\cos 27\theta} \]

### Step 5: Combine the terms
Now we can combine all three terms:
\[ b = \tan \theta \cdot \frac{1}{\cos 3\theta} + \tan 3\theta \cdot \frac{1}{\cos 9\theta} + \tan 9\theta \cdot \frac{1}{\cos 27\theta} \]

### Step 6: Factor out common terms
Notice that we can factor out \( \frac{1}{2} \) from the entire expression:
\[ b = \frac{1}{2} \left( \tan 3\theta - \tan \theta + \tan 9\theta - \tan 3\theta + \tan 27\theta - \tan 9\theta \right) \]

### Step 7: Simplify the expression
After simplifying, we find:
\[ b = \frac{1}{2} ( \tan 27\theta - \tan \theta ) \]

### Step 8: Relate \( b \) to \( a \)
Given that \( a = \tan 27\theta - \tan \theta \), we can substitute:
\[ b = \frac{1}{2} a \]

### Step 9: Final relationship
Thus, we can express \( a \) in terms of \( b \):
\[ a = 2b \]

### Conclusion
The final relationship between \( a \) and \( b \) is:
\[ \boxed{a = 2b} \]