Value of the expression `(sec1 1^(@)sec1 9^(@)-2cot7 1^(@))/tan11^(@)` is equal to

Options

A. `2 cot 11^(@)`
B. `tan 19^(@)`
C. `2 tan 11^(@)`
D. `(1)/(9)tan 19^(@)`

Explanation

To solve the expression \((\sec 11^\circ \sec 19^\circ - 2 \cot 71^\circ) / \tan 11^\circ\), we will break it down step by step.

### Step 1: Rewrite the trigonometric functions
We start with the expression:
\[ \frac{\sec 11^\circ \sec 19^\circ - 2 \cot 71^\circ}{\tan 11^\circ} \]
Recall that:
- \(\sec \theta = \frac{1}{\cos \theta}\)
- \(\cot \theta = \frac{\cos \theta}{\sin \theta}\)

So we can rewrite the expression as:
\[ \frac{\frac{1}{\cos 11^\circ \cos 19^\circ} - 2 \frac{\cos 71^\circ}{\sin 71^\circ}}{\tan 11^\circ} \]

### Step 2: Substitute cotangent
Next, we can substitute \(\cot 71^\circ\) using the identity \(\cot(90^\circ - \theta) = \tan \theta\):
\[ \cot 71^\circ = \tan(90^\circ - 71^\circ) = \tan 19^\circ \]
Thus, we can rewrite the expression as:
\[ \frac{\sec 11^\circ \sec 19^\circ - 2 \tan 19^\circ}{\tan 11^\circ} \]

### Step 3: Find a common denominator
Now, we need to find a common denominator for the terms in the numerator:
\[ \sec 11^\circ \sec 19^\circ = \frac{1}{\cos 11^\circ \cos 19^\circ} \]
So the numerator becomes:
\[ \frac{1 - 2 \tan 19^\circ \cos 11^\circ \cos 19^\circ}{\cos 11^\circ \cos 19^\circ} \]

### Step 4: Rewrite the numerator
Now we can rewrite the numerator:
\[ 1 - 2 \tan 19^\circ \cos 11^\circ \cos 19^\circ \]
Using the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we have:
\[ \tan 19^\circ = \frac{\sin 19^\circ}{\cos 19^\circ} \]
Thus, the expression simplifies to:
\[ 1 - 2 \frac{\sin 19^\circ \cos 11^\circ}{\cos 19^\circ} \]

### Step 5: Combine and simplify
Now, we can combine everything:
\[ \frac{1 - 2 \sin 19^\circ \cos 11^\circ}{\cos 11^\circ \cos 19^\circ} \]
Now, we can divide this by \(\tan 11^\circ\) which is \(\frac{\sin 11^\circ}{\cos 11^\circ}\):
\[ \frac{1 - 2 \sin 19^\circ \cos 11^\circ}{\cos 19^\circ} \cdot \frac{\cos 11^\circ}{\sin 11^\circ} \]

### Step 6: Final simplification
Now we can simplify:
\[ \frac{1 - 2 \sin 19^\circ \cos 11^\circ}{\sin 11^\circ \cos 19^\circ} \]
Using the sine subtraction formula:
\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]
We can express this as:
\[ \frac{\sin 30^\circ - \sin 8^\circ}{\sin 11^\circ \cos 19^\circ} \]
Using the sine difference identity, we find:
\[ \frac{2 \sin(30^\circ - 8^\circ) \cos(30^\circ + 8^\circ)}{\sin 11^\circ \cos 19^\circ} \]

### Step 7: Evaluate the expression
After simplification, we find that:
\[ \frac{2 \tan 11^\circ}{\tan 11^\circ} = 2 \]

Thus, the value of the expression is:
\[ \boxed{2} \]