`"sin"^(2)(pi)/(6)+"cos"^(2)(pi)/(3)-"tan"^(2)(pi)/(4)+(1)/(2)`is:

Options

A. `(1)/(2)`
B. `-(1)/(2)`
C. 1
D. 0

Answer

To solve the expression \( \sin^2\left(\frac{\pi}{6}\right) + \cos^2\left(\frac{\pi}{3}\right) - \tan^2\left(\frac{\pi}{4}\right) + \frac{1}{2} \), we will calculate each term step by step.

### Step 1: Calculate \( \sin^2\left(\frac{\pi}{6}\right) \)
We know that:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Thus,
\[ \sin^2\left(\frac{\pi}{6}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 2: Calculate \( \cos^2\left(\frac{\pi}{3}\right) \)
We know that:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]
Thus,
\[ \cos^2\left(\frac{\pi}{3}\right) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

### Step 3: Calculate \( \tan^2\left(\frac{\pi}{4}\right) \)
We know that:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
Thus,
\[ \tan^2\left(\frac{\pi}{4}\right) = 1^2 = 1 \]

### Step 4: Substitute the values into the expression
Now, substituting the calculated values into the original expression:
\[ \sin^2\left(\frac{\pi}{6}\right) + \cos^2\left(\frac{\pi}{3}\right) - \tan^2\left(\frac{\pi}{4}\right) + \frac{1}{2} \]
becomes:
\[ \frac{1}{4} + \frac{1}{4} - 1 + \frac{1}{2} \]

### Step 5: Simplify the expression
Now, let's simplify:
\[ \frac{1}{4} + \frac{1}{4} = \frac{2}{4} = \frac{1}{2} \]
So the expression now is:
\[ \frac{1}{2} - 1 + \frac{1}{2} \]
This simplifies to:
\[ \frac{1}{2} + \frac{1}{2} - 1 = 1 - 1 = 0 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0} \]