The value of x, if `x cos ^(2) 45 ^(@) - sec ^(2) 60^(@) + sin ^(2) 30^(@) = (1)/(8),` is

Options

A. `31`
B. 32
C. `(31)/(4)`
D. None of these

Answer

To solve the equation \( x \cos^2 45^\circ - \sec^2 60^\circ + \sin^2 30^\circ = \frac{1}{8} \), we will follow these steps:

### Step 1: Substitute the trigonometric values
We know the following trigonometric values:
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sec 60^\circ = 2 \) (since \( \sec \theta = \frac{1}{\cos \theta} \) and \( \cos 60^\circ = \frac{1}{2} \))
- \( \sin 30^\circ = \frac{1}{2} \)

Now, substituting these values into the equation:
\[ x \left( \frac{1}{\sqrt{2}} \right)^2 - (2)^2 + \left( \frac{1}{2} \right)^2 = \frac{1}{8} \]

### Step 2: Simplify the equation
Calculating the squares:
\[ x \left( \frac{1}{2} \right) - 4 + \frac{1}{4} = \frac{1}{8} \]

This simplifies to:
\[ \frac{x}{2} - 4 + \frac{1}{4} = \frac{1}{8} \]

### Step 3: Combine like terms
To combine the constants, we need a common denominator. The least common multiple of 1, 4, and 8 is 8. Rewriting the constants:
\[ \frac{x}{2} - \frac{32}{8} + \frac{2}{8} = \frac{1}{8} \]
This simplifies to:
\[ \frac{x}{2} - \frac{30}{8} = \frac{1}{8} \]

### Step 4: Isolate \( x \)
Now, add \( \frac{30}{8} \) to both sides:
\[ \frac{x}{2} = \frac{1}{8} + \frac{30}{8} \]
\[ \frac{x}{2} = \frac{31}{8} \]

### Step 5: Solve for \( x \)
Multiply both sides by 2 to solve for \( x \):
\[ x = 2 \cdot \frac{31}{8} = \frac{62}{8} = \frac{31}{4} \]

Thus, the value of \( x \) is \( \frac{31}{4} \).

### Final Answer
\[ \boxed{\frac{31}{4}} \]