If `sin 7x = cos 11x , 0^(@) lt x lt 90^(@)`, then the value of `tan 9x` is:

Options

A. `( sqrt3)/( 2)`
B. 1
C. `(1)/( sqrt3)`
D. `sqrt3`

Answer

To solve the equation \( \sin 7x = \cos 11x \) for \( 0^\circ < x < 90^\circ \) and find the value of \( \tan 9x \), we can follow these steps:

### Step 1: Use the co-function identity
We know that \( \cos \theta = \sin(90^\circ - \theta) \). Therefore, we can rewrite the equation:
\[ \sin 7x = \cos 11x \implies \sin 7x = \sin(90^\circ - 11x) \]

### Step 2: Set the arguments equal
Since the sine function is equal for two angles, we can set the arguments equal to each other:
\[ 7x = 90^\circ - 11x \]

### Step 3: Solve for \( x \)
Now, we will solve for \( x \):
\[ 7x + 11x = 90^\circ \]
\[ 18x = 90^\circ \]
\[ x = \frac{90^\circ}{18} = 5^\circ \]

### Step 4: Find \( 9x \)
Now that we have \( x \), we can find \( 9x \):
\[ 9x = 9 \times 5^\circ = 45^\circ \]

### Step 5: Calculate \( \tan 9x \)
Now we can find \( \tan 9x \):
\[ \tan 9x = \tan 45^\circ = 1 \]

### Final Answer
Thus, the value of \( \tan 9x \) is:
\[ \boxed{1} \]