Solve for x :
`cos^(2)30^(@)+cos^(2)x=1`

Answer

To solve the equation \( \cos^2(30^\circ) + \cos^2(x) = 1 \), we can follow these steps:

### Step 1: Calculate \( \cos(30^\circ) \)
We know that:
\[ \cos(30^\circ) = \frac{\sqrt{3}}{2} \]

### Step 2: Substitute \( \cos(30^\circ) \) into the equation
Now we substitute \( \cos(30^\circ) \) into the equation:
\[ \cos^2(30^\circ) + \cos^2(x) = 1 \]
This becomes:
\[ \left(\frac{\sqrt{3}}{2}\right)^2 + \cos^2(x) = 1 \]

### Step 3: Simplify \( \cos^2(30^\circ) \)
Calculating \( \left(\frac{\sqrt{3}}{2}\right)^2 \):
\[ \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]
So the equation now is:
\[ \frac{3}{4} + \cos^2(x) = 1 \]

### Step 4: Isolate \( \cos^2(x) \)
Subtract \( \frac{3}{4} \) from both sides:
\[ \cos^2(x) = 1 - \frac{3}{4} \]
This simplifies to:
\[ \cos^2(x) = \frac{1}{4} \]

### Step 5: Take the square root of both sides
Taking the square root gives us:
\[ \cos(x) = \pm \frac{1}{2} \]

### Step 6: Find the angles for \( \cos(x) = \frac{1}{2} \) and \( \cos(x) = -\frac{1}{2} \)
The angles where \( \cos(x) = \frac{1}{2} \) are:
\[ x = 60^\circ \quad \text{and} \quad x = 300^\circ \quad (\text{since } 300^\circ = 360^\circ - 60^\circ) \]
The angle where \( \cos(x) = -\frac{1}{2} \) is:
\[ x = 120^\circ \quad \text{and} \quad x = 240^\circ \]

### Final Answer
Thus, the solutions for \( x \) are:
\[ x = 60^\circ, 120^\circ, 240^\circ, 300^\circ \]