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

Options

A. `(tan 2A)/( tan 8A)`
B. `(tan 8A)/(tan 2A)`
C. `(cot 8A)/(cot 2A)`
D. `(tan 6A)/(tan 2A)`

Answer

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

### Step 1: Rewrite Secant in Terms of Cosine
Recall that \(\sec x = \frac{1}{\cos x}\). Therefore, we can rewrite the expression as follows:

\[ \sec 8A - 1 = \frac{1}{\cos 8A} - 1 = \frac{1 - \cos 8A}{\cos 8A} \]

\[ \sec 4A - 1 = \frac{1}{\cos 4A} - 1 = \frac{1 - \cos 4A}{\cos 4A} \]

### Step 2: Substitute Back into the Expression
Now substitute these back into the original expression:

\[ \frac{\sec 8A - 1}{\sec 4A - 1} = \frac{\frac{1 - \cos 8A}{\cos 8A}}{\frac{1 - \cos 4A}{\cos 4A}} = \frac{(1 - \cos 8A) \cdot \cos 4A}{(1 - \cos 4A) \cdot \cos 8A} \]

### Step 3: Simplify the Expression
Now, we can simplify the expression. The cosine terms can be canceled if they are not equal to zero:

\[ = \frac{(1 - \cos 8A) \cdot \cos 4A}{(1 - \cos 4A) \cdot \cos 8A} \]

### Step 4: Use the Identity for Cosine
Using the identity \(1 - \cos x = 2 \sin^2 \left(\frac{x}{2}\right)\), we can rewrite both \(1 - \cos 8A\) and \(1 - \cos 4A\):

\[ 1 - \cos 8A = 2 \sin^2(4A) \]
\[ 1 - \cos 4A = 2 \sin^2(2A) \]

### Step 5: Substitute the Identities
Substituting these identities into our expression gives:

\[ = \frac{2 \sin^2(4A) \cdot \cos 4A}{2 \sin^2(2A) \cdot \cos 8A} \]

### Step 6: Cancel Out the Common Factor
The factor of \(2\) cancels out:

\[ = \frac{\sin^2(4A) \cdot \cos 4A}{\sin^2(2A) \cdot \cos 8A} \]

### Step 7: Use the Double Angle Identity
Recall that \(\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\). Thus, we can express \(\sin(4A)\) as:

\[ \sin(4A) = 2 \sin(2A) \cos(2A) \]

Substituting this back into our expression:

\[ = \frac{(2 \sin(2A) \cos(2A))^2 \cdot \cos 4A}{\sin^2(2A) \cdot \cos 8A} \]

### Step 8: Final Simplification
This simplifies to:

\[ = \frac{4 \sin^2(2A) \cos^2(2A) \cdot \cos 4A}{\sin^2(2A) \cdot \cos 8A} \]

Now, cancel \(\sin^2(2A)\):

\[ = \frac{4 \cos^2(2A) \cdot \cos 4A}{\cos 8A} \]

### Final Answer
Thus, the final simplified expression is:

\[ \frac{4 \cos^2(2A) \cdot \cos 4A}{\cos 8A} \]