value of `(5sin75^(@)sin77^(@)+2cos13^(@)cos15^(@))/(cos15^(@)sin77^(@))-(7sin81^(@))/(cos9^(@))` is

Options

A. `-1`
B. `0`
C. `1`
D. `2`

Answer

To solve the expression \((5\sin 75^\circ \sin 77^\circ + 2\cos 13^\circ \cos 15^\circ) / (\cos 15^\circ \sin 77^\circ) - (7\sin 81^\circ / \cos 9^\circ)\), we will follow these steps:

### Step 1: Use Trigonometric Identities
We know that:
- \(\sin(90^\circ - \theta) = \cos(\theta)\)
- \(\cos(90^\circ - \theta) = \sin(\theta)\)

### Step 2: Rewrite the Sine and Cosine Functions
We can rewrite the angles in the expression using the identities:
- \(\sin 75^\circ = \sin(90^\circ - 15^\circ) = \cos 15^\circ\)
- \(\sin 77^\circ = \sin(90^\circ - 13^\circ) = \cos 13^\circ\)
- \(\sin 81^\circ = \sin(90^\circ - 9^\circ) = \cos 9^\circ\)

### Step 3: Substitute the Values
Substituting these values into the expression gives:
\[ = \frac{5\cos 15^\circ \cos 13^\circ + 2\cos 13^\circ \cos 15^\circ}{\cos 15^\circ \cos 13^\circ} - \frac{7\cos 9^\circ}{\cos 9^\circ} \]

### Step 4: Simplify the Expression
Now, we can simplify the expression:
\[ = \frac{(5 + 2)\cos 15^\circ \cos 13^\circ}{\cos 15^\circ \cos 13^\circ} - 7 \]
\[ = 7 - 7 \]

### Step 5: Final Result
This simplifies to:
\[ = 0 \]

Thus, the value of the given expression is **0**.