If `cos^(2)30^(@)+cos^(2)45^(@)+cos^(2)60^(@)=x`, then find the value of 'x'

Answer

To find the value of \( x \) in the equation \( \cos^2 30^\circ + \cos^2 45^\circ + \cos^2 60^\circ = x \), we will calculate each term step by step.

### Step 1: Calculate \( \cos^2 30^\circ \)
We know that:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \]
Now, squaring this value:
\[ \cos^2 30^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

### Step 2: Calculate \( \cos^2 45^\circ \)
We know that:
\[ \cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \]
Now, squaring this value:
\[ \cos^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \]

### Step 3: Calculate \( \cos^2 60^\circ \)
We know that:
\[ \cos 60^\circ = \frac{1}{2} \]
Now, squaring this value:
\[ \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 4: Sum the values
Now we can sum all the calculated values:
\[ x = \cos^2 30^\circ + \cos^2 45^\circ + \cos^2 60^\circ \]
Substituting the values we found:
\[ x = \frac{3}{4} + \frac{1}{2} + \frac{1}{4} \]

### Step 5: Find a common denominator
The common denominator for \( 4 \) and \( 2 \) is \( 4 \). We can rewrite \( \frac{1}{2} \) as \( \frac{2}{4} \):
\[ x = \frac{3}{4} + \frac{2}{4} + \frac{1}{4} \]

### Step 6: Combine the fractions
Now, we can combine the fractions:
\[ x = \frac{3 + 2 + 1}{4} = \frac{6}{4} \]

### Step 7: Simplify the result
Finally, simplifying \( \frac{6}{4} \):
\[ x = \frac{3}{2} \]

Thus, the value of \( x \) is:
\[ \boxed{\frac{3}{2}} \]