`( sec 8 theta -1 )/( sec 4 theta -1) = `

Options

A. `tan 8 theta tan 2 theta`
B. `tan 8 theta // tan 2 theta`
C. `cot 8 theta cot 2 theta`
D. none

Answer

To solve the expression \(( \sec 8\theta - 1 ) / ( \sec 4\theta - 1 )\), we will follow these steps:

### Step 1: Rewrite secant in terms of cosine
Recall that \(\sec \theta = \frac{1}{\cos \theta}\). Therefore, we can rewrite the expression:
\[ \sec 8\theta - 1 = \frac{1}{\cos 8\theta} - 1 = \frac{1 - \cos 8\theta}{\cos 8\theta} \]
\[ \sec 4\theta - 1 = \frac{1}{\cos 4\theta} - 1 = \frac{1 - \cos 4\theta}{\cos 4\theta} \]

### Step 2: Substitute into the original expression
Substituting these into the original expression gives:
\[ \frac{\sec 8\theta - 1}{\sec 4\theta - 1} = \frac{\frac{1 - \cos 8\theta}{\cos 8\theta}}{\frac{1 - \cos 4\theta}{\cos 4\theta}} = \frac{(1 - \cos 8\theta) \cdot \cos 4\theta}{(1 - \cos 4\theta) \cdot \cos 8\theta} \]

### Step 3: Use the trigonometric identity
Using the identity \(1 - \cos x = 2 \sin^2 \left(\frac{x}{2}\right)\), we can express \(1 - \cos 8\theta\) and \(1 - \cos 4\theta\):
\[ 1 - \cos 8\theta = 2 \sin^2 (4\theta) \]
\[ 1 - \cos 4\theta = 2 \sin^2 (2\theta) \]

### Step 4: Substitute the identities back into the expression
Substituting these identities into our expression gives:
\[ \frac{(2 \sin^2 (4\theta)) \cdot \cos 4\theta}{(2 \sin^2 (2\theta)) \cdot \cos 8\theta} \]

### Step 5: Simplify the expression
The \(2\) cancels out:
\[ \frac{\sin^2 (4\theta) \cdot \cos 4\theta}{\sin^2 (2\theta) \cdot \cos 8\theta} \]

### Step 6: Use the double angle identity
We can use the double angle identity for sine:
\[ \sin (2\theta) = 2 \sin \theta \cos \theta \]
Thus, \(\sin^2 (4\theta) = (2 \sin (2\theta) \cos (2\theta))^2 = 4 \sin^2 (2\theta) \cos^2 (2\theta)\).

### Step 7: Substitute and simplify further
Substituting this back:
\[ \frac{4 \sin^2 (2\theta) \cos^2 (2\theta) \cdot \cos 4\theta}{\sin^2 (2\theta) \cdot \cos 8\theta} \]
Cancelling \(\sin^2 (2\theta)\):
\[ \frac{4 \cos^2 (2\theta) \cdot \cos 4\theta}{\cos 8\theta} \]

### Step 8: Final simplification
Using the identity \(\cos 8\theta = 2 \cos^2 (4\theta) - 1\), we can express \(\cos 8\theta\) in terms of \(\cos 4\theta\) and simplify further if needed.

### Final Result
The final expression simplifies to:
\[ \tan 8\theta / \tan 2\theta \]