If `11 x` is an acute angle and tan 11 x = cot 7 x , then what is the value of `x` ?

Options

A. `5^(@)`
B. `6^(@)`
C. `7^(@)`
D. `8^(@)`

Answer

To solve the equation \( \tan(11x) = \cot(7x) \), we can follow these steps:

### Step 1: Use the cotangent identity
We know that \( \cot(θ) = \frac{1}{\tan(θ)} \). Therefore, we can rewrite the equation:
\[ \tan(11x) = \frac{1}{\tan(7x)} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ \tan(11x) \cdot \tan(7x) = 1 \]

### Step 3: Use the tangent product identity
From the identity \( \tan(A) \tan(B) = 1 \) when \( A + B = 90^\circ \), we can deduce:
\[ 11x + 7x = 90^\circ \]

### Step 4: Combine like terms
Combining the terms gives us:
\[ 18x = 90^\circ \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \) by dividing both sides by 18:
\[ x = \frac{90^\circ}{18} = 5^\circ \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{5^\circ} \]

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