Find the value of
`cos""(pi)/(12)(sin""(5pi)/(12)+cos""(pi)/(4))+sin""(pi)/(12)(cos""(5pi)/(12)-sin""(pi)/(4))`.

Explanation

To solve the expression

\[ \cos\left(\frac{\pi}{12}\right) \left(\sin\left(\frac{5\pi}{12}\right) + \cos\left(\frac{\pi}{4}\right)\right) + \sin\left(\frac{\pi}{12}\right) \left(\cos\left(\frac{5\pi}{12}\right) - \sin\left(\frac{\pi}{4}\right)\right), \]

we will follow these steps:

### Step 1: Convert radians to degrees
First, we convert the angles from radians to degrees for easier calculation:
- \(\frac{\pi}{12} = 15^\circ\)
- \(\frac{5\pi}{12} = 75^\circ\)
- \(\frac{\pi}{4} = 45^\circ\)

### Step 2: Rewrite the expression
Now we can rewrite the expression using degrees:

\[ \cos(15^\circ) \left(\sin(75^\circ) + \cos(45^\circ)\right) + \sin(15^\circ) \left(\cos(75^\circ) - \sin(45^\circ)\right) \]

### Step 3: Calculate the trigonometric values
Now we calculate the trigonometric values:
- \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)
- \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)
- \(\sin(75^\circ) = \cos(15^\circ)\) (since \(\sin(75^\circ) = \sin(90^\circ - 15^\circ)\))
- \(\cos(75^\circ) = \sin(15^\circ)\)

### Step 4: Substitute values
Substituting these values back into the expression gives us:

\[ \cos(15^\circ) \left(\cos(15^\circ) + \frac{\sqrt{2}}{2}\right) + \sin(15^\circ) \left(\sin(15^\circ) - \frac{\sqrt{2}}{2}\right) \]

### Step 5: Expand the expression
Now we expand the expression:

\[ \cos^2(15^\circ) + \frac{\sqrt{2}}{2} \cos(15^\circ) + \sin^2(15^\circ) - \frac{\sqrt{2}}{2} \sin(15^\circ) \]

### Step 6: Use the Pythagorean identity
Using the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\):

\[ 1 + \frac{\sqrt{2}}{2} \cos(15^\circ) - \frac{\sqrt{2}}{2} \sin(15^\circ) \]

### Step 7: Factor out \(\frac{\sqrt{2}}{2}\)
Now we can factor out \(\frac{\sqrt{2}}{2}\):

\[ 1 + \frac{\sqrt{2}}{2} \left(\cos(15^\circ) - \sin(15^\circ)\right) \]

### Step 8: Calculate the final value
To find the final value, we need to evaluate \(\cos(15^\circ) - \sin(15^\circ)\).

Using the values of \(\cos(15^\circ)\) and \(\sin(15^\circ)\), we can compute this value, but for simplicity, we can directly compute the numerical approximation or use a calculator to find \(1 + \frac{\sqrt{2}}{2} \left(\cos(15^\circ) - \sin(15^\circ)\right)\).

### Final Result
After evaluating, we find that the final value of the expression is:

\[ \frac{3}{2} \]