`cos51^(@)-sin39^(@)+sin37^(@)-cos53^(@)=?`

Options

A. 0
B. 1
C. `cos104^(@)`
D. `sin39^(@)`

Answer

To solve the equation \( \cos 51^\circ - \sin 39^\circ + \sin 37^\circ - \cos 53^\circ = ? \), we can use the trigonometric identities.

### Step 1: Rewrite the terms using co-function identities

Recall the co-function identities:
- \( \cos(90^\circ - \theta) = \sin(\theta) \)
- \( \sin(90^\circ - \theta) = \cos(\theta) \)

Using these identities, we can rewrite \( \cos 51^\circ \) and \( \cos 53^\circ \):
- \( \cos 51^\circ = \sin(90^\circ - 51^\circ) = \sin 39^\circ \)
- \( \cos 53^\circ = \sin(90^\circ - 53^\circ) = \sin 37^\circ \)

### Step 2: Substitute the rewritten terms into the equation

Now, substitute these identities back into the original equation:
\[ \cos 51^\circ - \sin 39^\circ + \sin 37^\circ - \cos 53^\circ = \sin 39^\circ - \sin 39^\circ + \sin 37^\circ - \sin 37^\circ \]

### Step 3: Simplify the equation

Now, simplify the equation:
\[ \sin 39^\circ - \sin 39^\circ + \sin 37^\circ - \sin 37^\circ = 0 + 0 = 0 \]

### Final Answer

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

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