`(Cos^(2)phi+(1)/(cosec^(2)phi))+17=x` . What is the value of `x^(2)` ?

Options

A. `18`
B. `324`
C. `256`
D. `16`

Answer

To solve the equation \( \cos^2 \phi + \frac{1}{\csc^2 \phi} + 17 = x \) and find the value of \( x^2 \), we can follow these steps:

### Step 1: Rewrite the equation
We know that \( \csc^2 \phi = \frac{1}{\sin^2 \phi} \). Therefore, we can rewrite \( \frac{1}{\csc^2 \phi} \) as \( \sin^2 \phi \):
\[ \cos^2 \phi + \sin^2 \phi + 17 = x \]

### Step 2: Use the Pythagorean identity
Using the Pythagorean identity, we know that:
\[ \cos^2 \phi + \sin^2 \phi = 1 \]
Substituting this into our equation gives:
\[ 1 + 17 = x \]

### Step 3: Simplify the equation
Now, simplify the left side:
\[ 18 = x \]

### Step 4: Calculate \( x^2 \)
Now that we have found \( x \), we can calculate \( x^2 \):
\[ x^2 = 18^2 = 324 \]

### Final Answer
Thus, the value of \( x^2 \) is:
\[ \boxed{324} \]