The value of `(1)/(sqrt(2))"sin"(pi)/(6)"cos"(pi)/(4)-"cot"(pi)/(3)"sec"(pi)/(6)+(5"tan"(pi)/(4))/("12sin"(pi)/(2))` is equal to

Options

A. 0
B. 1
C. 2
D. `(3)/(2)`

Answer

To solve the expression

\[ \frac{1}{\sqrt{2}} \sin\left(\frac{\pi}{6}\right) \cos\left(\frac{\pi}{4}\right) - \cot\left(\frac{\pi}{3}\right) \sec\left(\frac{\pi}{6}\right) + \frac{5 \tan\left(\frac{\pi}{4}\right)}{12 \sin\left(\frac{\pi}{2}\right)} \]

we will evaluate each trigonometric function step by step.

### Step 1: Evaluate each trigonometric function

1. **Evaluate \(\sin\left(\frac{\pi}{6}\right)\)**:
\[ \sin\left(\frac{\pi}{6}\right) = \sin(30^\circ) = \frac{1}{2} \]

2. **Evaluate \(\cos\left(\frac{\pi}{4}\right)\)**:
\[ \cos\left(\frac{\pi}{4}\right) = \cos(45^\circ) = \frac{1}{\sqrt{2}} \]

3. **Evaluate \(\cot\left(\frac{\pi}{3}\right)\)**:
\[ \cot\left(\frac{\pi}{3}\right) = \cot(60^\circ) = \frac{1}{\sqrt{3}} \]

4. **Evaluate \(\sec\left(\frac{\pi}{6}\right)\)**:
\[ \sec\left(\frac{\pi}{6}\right) = \sec(30^\circ) = \frac{2}{\sqrt{3}} \]

5. **Evaluate \(\tan\left(\frac{\pi}{4}\right)\)**:
\[ \tan\left(\frac{\pi}{4}\right) = \tan(45^\circ) = 1 \]

6. **Evaluate \(\sin\left(\frac{\pi}{2}\right)\)**:
\[ \sin\left(\frac{\pi}{2}\right) = \sin(90^\circ) = 1 \]

### Step 2: Substitute the values into the expression

Now we substitute these values back into the expression:

\[ \frac{1}{\sqrt{2}} \cdot \frac{1}{2} \cdot \frac{1}{\sqrt{2}} - \left(\frac{1}{\sqrt{3}} \cdot \frac{2}{\sqrt{3}}\right) + \frac{5 \cdot 1}{12 \cdot 1} \]

### Step 3: Simplify each term

1. **First term**:
\[ \frac{1}{\sqrt{2}} \cdot \frac{1}{2} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{1}{4} \]

2. **Second term**:
\[ \frac{1}{\sqrt{3}} \cdot \frac{2}{\sqrt{3}} = \frac{2}{3} \]

3. **Third term**:
\[ \frac{5 \cdot 1}{12 \cdot 1} = \frac{5}{12} \]

### Step 4: Combine the terms

Now we combine all the terms:

\[ \frac{1}{4} - \frac{2}{3} + \frac{5}{12} \]

### Step 5: Find a common denominator

The common denominator for \(4\), \(3\), and \(12\) is \(12\). We convert each fraction:

1. **Convert \(\frac{1}{4}\)**:
\[ \frac{1}{4} = \frac{3}{12} \]

2. **Convert \(\frac{2}{3}\)**:
\[ \frac{2}{3} = \frac{8}{12} \]

3. **Already in terms of \(12\)**:
\[ \frac{5}{12} \]

### Step 6: Combine the fractions

Now we can combine them:

\[ \frac{3}{12} - \frac{8}{12} + \frac{5}{12} = \frac{3 - 8 + 5}{12} = \frac{0}{12} = 0 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{0} \]