Find the values of `"sin"34^(@) + cos 64^(@) - cos4^(@) `.

Answer

To solve the expression \( \sin 34^\circ + \cos 64^\circ - \cos 4^\circ \), we can follow these steps:

### Step 1: Rewrite \( \cos 64^\circ \) and \( \cos 4^\circ \)
We know that:
- \( \cos 64^\circ = \cos(90^\circ - 26^\circ) = \sin 26^\circ \)
- \( \cos 4^\circ = \cos(34^\circ - 30^\circ) \)

Using the cosine subtraction formula:
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
we can write:
\[ \cos 4^\circ = \cos 34^\circ \cos 30^\circ + \sin 34^\circ \sin 30^\circ \]

### Step 2: Substitute values
Now substituting the known values:
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 30^\circ = \frac{1}{2} \)

Thus, we have:
\[ \cos 4^\circ = \cos 34^\circ \cdot \frac{\sqrt{3}}{2} + \sin 34^\circ \cdot \frac{1}{2} \]

### Step 3: Substitute back into the original expression
Now substituting \( \cos 64^\circ \) and \( \cos 4^\circ \) back into the original expression:
\[ \sin 34^\circ + \sin 26^\circ - \left( \cos 34^\circ \cdot \frac{\sqrt{3}}{2} + \sin 34^\circ \cdot \frac{1}{2} \right) \]

### Step 4: Simplify the expression
Now we can simplify:
\[ \sin 34^\circ + \sin 26^\circ - \left( \cos 34^\circ \cdot \frac{\sqrt{3}}{2} + \frac{1}{2} \sin 34^\circ \right) \]
This simplifies to:
\[ \sin 34^\circ - \frac{1}{2} \sin 34^\circ + \sin 26^\circ - \frac{\sqrt{3}}{2} \cos 34^\circ \]
\[ = \frac{1}{2} \sin 34^\circ + \sin 26^\circ - \frac{\sqrt{3}}{2} \cos 34^\circ \]

### Step 5: Use the sine and cosine identity
We can use the identity \( \sin x + \cos x = \sqrt{2} \sin(x + 45^\circ) \) to further simplify if necessary, but we can also evaluate the expression directly.

### Final Calculation
After evaluating the expression, we find that:
\[ \sin 34^\circ + \cos 64^\circ - \cos 4^\circ = 0 \]

Thus, the value of \( \sin 34^\circ + \cos 64^\circ - \cos 4^\circ \) is \( 0 \).