What is the value of
`(sin34^@cos236^@-sin56^@sin124^@)/(cos28^@cos88^@+cos178^@sin208^@)`

Options

A. `-2`
B. `-1`
C. 2
D. 1

Answer

To solve the expression

\[ \frac{\sin 34^\circ \cos 236^\circ - \sin 56^\circ \sin 124^\circ}{\cos 28^\circ \cos 88^\circ + \cos 178^\circ \sin 208^\circ} \]

we will simplify both the numerator and the denominator step by step.

### Step 1: Simplify the numerator

First, we simplify the term \(\cos 236^\circ\) and \(\sin 124^\circ\):
- \(\cos 236^\circ = \cos(180^\circ + 56^\circ) = -\cos 56^\circ\)
- \(\sin 124^\circ = \sin(180^\circ - 56^\circ) = \sin 56^\circ\)

Now substitute these values into the numerator:

\[ \sin 34^\circ \cos 236^\circ - \sin 56^\circ \sin 124^\circ = \sin 34^\circ (-\cos 56^\circ) - \sin 56^\circ \sin 56^\circ \]

This simplifies to:

\[ -\sin 34^\circ \cos 56^\circ - \sin^2 56^\circ \]

### Step 2: Factor the numerator

We can factor out \(-1\):

\[ -(\sin 34^\circ \cos 56^\circ + \sin^2 56^\circ) \]

### Step 3: Simplify the denominator

Now we simplify the denominator:

- \(\cos 178^\circ = \cos(180^\circ - 2^\circ) = -\cos 2^\circ\)
- \(\sin 208^\circ = \sin(180^\circ + 28^\circ) = -\sin 28^\circ\)

Now substitute these values into the denominator:

\[ \cos 28^\circ \cos 88^\circ + \cos 178^\circ \sin 208^\circ = \cos 28^\circ \cos 88^\circ + (-\cos 2^\circ)(-\sin 28^\circ) \]

This simplifies to:

\[ \cos 28^\circ \cos 88^\circ + \cos 2^\circ \sin 28^\circ \]

### Step 4: Apply trigonometric identities

Using the identity \(\sin(a + b) = \sin a \cos b + \cos a \sin b\) on the numerator:

\[ \sin 34^\circ \cos 56^\circ + \sin^2 56^\circ = \sin(34^\circ + 56^\circ) = \sin 90^\circ = 1 \]

Thus, the numerator becomes:

\[ -(1) = -1 \]

For the denominator, we can use the identity \(\cos a \cos b + \sin a \sin b = \cos(a - b)\):

\[ \cos 28^\circ \cos 88^\circ + \sin 28^\circ \cos 2^\circ = \cos(28^\circ - 88^\circ) = \cos(-60^\circ) = \cos 60^\circ = \frac{1}{2} \]

### Step 5: Final expression

Now substituting back into the original expression:

\[ \frac{-1}{\frac{1}{2}} = -2 \]

### Conclusion

Thus, the value of the expression is:

\[ \boxed{-2} \]