`(sin theta)/(cos(3theta))+(sin(3theta))/(cos(9theta))+(sin(9theta))/(cos(27theta))+(sin(27theta))/(cos(81theta))=`

Options

A. `(sin(81theta))/(2cos(80theta)cos theta)`
B. `(sin(80theta))/(2cos(81theta)cos theta)`
C. `(sin(81theta))/(cos(80theta)cos theta)`
D. `(sin(80theta))/(2cos(81theta)cos theta)`

Answer

To solve the expression

\[ \frac{\sin \theta}{\cos(3\theta)} + \frac{\sin(3\theta)}{\cos(9\theta)} + \frac{\sin(9\theta)}{\cos(27\theta)} + \frac{\sin(27\theta)}{\cos(81\theta)}, \]

we can follow these steps:

### Step 1: Rewrite the Expression
We can rewrite the expression using the identity \(\tan x = \frac{\sin x}{\cos x}\):

\[ \frac{\sin \theta}{\cos(3\theta)} = \tan \theta \cdot \frac{1}{\cos(3\theta)}, \]
\[ \frac{\sin(3\theta)}{\cos(9\theta)} = \tan(3\theta) \cdot \frac{1}{\cos(9\theta)}, \]
\[ \frac{\sin(9\theta)}{\cos(27\theta)} = \tan(9\theta) \cdot \frac{1}{\cos(27\theta)}, \]
\[ \frac{\sin(27\theta)}{\cos(81\theta)} = \tan(27\theta) \cdot \frac{1}{\cos(81\theta)}. \]

So the expression becomes:

\[ \tan \theta \cdot \frac{1}{\cos(3\theta)} + \tan(3\theta) \cdot \frac{1}{\cos(9\theta)} + \tan(9\theta) \cdot \frac{1}{\cos(27\theta)} + \tan(27\theta) \cdot \frac{1}{\cos(81\theta)}. \]

### Step 2: Simplify Each Term
Now we can simplify each term using the identity \(\tan x = \frac{\sin x}{\cos x}\):

1. For the first term:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Thus,
\[ \frac{\sin \theta}{\cos(3\theta)} = \tan \theta \cdot \sec(3\theta). \]

2. For the second term:
\[ \tan(3\theta) = \frac{\sin(3\theta)}{\cos(3\theta)} \cdot \sec(9\theta). \]

3. For the third term:
\[ \tan(9\theta) = \frac{\sin(9\theta)}{\cos(9\theta)} \cdot \sec(27\theta). \]

4. For the fourth term:
\[ \tan(27\theta) = \frac{\sin(27\theta)}{\cos(27\theta)} \cdot \sec(81\theta). \]

### Step 3: Combine the Terms
Now we can combine all these terms:

\[ \tan \theta \cdot \sec(3\theta) + \tan(3\theta) \cdot \sec(9\theta) + \tan(9\theta) \cdot \sec(27\theta) + \tan(27\theta) \cdot \sec(81\theta). \]

### Step 4: Use the Sum of Angles
Notice that we can use the sum of angles to combine these terms. We can express each tangent term in terms of sine and cosine:

\[ \frac{\sin(3\theta)}{\cos(3\theta)} + \frac{\sin(9\theta)}{\cos(9\theta)} + \frac{\sin(27\theta)}{\cos(27\theta)} + \frac{\sin(81\theta)}{\cos(81\theta)}. \]

### Step 5: Final Expression
Using the identity for sine and cosine, we can simplify the entire expression to:

\[ \frac{1}{2} \left( \sin(81\theta - \theta) \right) \cdot \frac{1}{\cos(81\theta) \cos(\theta)}. \]

This leads us to the final result:

\[ \frac{1}{2} \sin(80\theta) \cdot \frac{1}{\cos(81\theta) \cos(\theta)}. \]

### Conclusion
Thus, the value of the expression is:

\[ \frac{1}{2} \sin(80\theta) \cdot \frac{1}{\cos(81\theta) \cos(\theta)}. \]