The value `((cos9^(@)+sin81^(@))(sec9^(@)+cosec81^(@)))/(sin56^(@) sec34^(@)+cos25^(@)cosec65^(@))` is :

Options

A. `(1)/(2)`
B. 4
C. `(1)/(4)`
D. 2

Answer

To solve the expression

\[ \frac{(\cos 9^\circ + \sin 81^\circ)(\sec 9^\circ + \csc 81^\circ)}{(\sin 56^\circ \sec 34^\circ + \cos 25^\circ \csc 65^\circ)}, \]

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

### Step 1: Simplifying the Numerator

1. **Recognize Trigonometric Identities**:
We know that \(\sin(90^\circ - \theta) = \cos(\theta)\) and \(\cos(90^\circ - \theta) = \sin(\theta)\).

Thus, we can rewrite \(\sin 81^\circ\) as \(\cos 9^\circ\) since \(81^\circ = 90^\circ - 9^\circ\).

Therefore,
\[ \cos 9^\circ + \sin 81^\circ = \cos 9^\circ + \cos 9^\circ = 2 \cos 9^\circ. \]

2. **Simplifying the Second Part**:
For \(\sec 9^\circ + \csc 81^\circ\), we can rewrite \(\csc 81^\circ\) as \(\sec 9^\circ\) using the same identity:
\[ \sec 9^\circ + \csc 81^\circ = \sec 9^\circ + \sec 9^\circ = 2 \sec 9^\circ. \]

3. **Combining the Parts**:
Now, substituting back into the numerator:
\[ (\cos 9^\circ + \sin 81^\circ)(\sec 9^\circ + \csc 81^\circ) = (2 \cos 9^\circ)(2 \sec 9^\circ) = 4 \cos 9^\circ \sec 9^\circ. \]
Since \(\cos 9^\circ \sec 9^\circ = 1\), we have:
\[ 4 \cos 9^\circ \sec 9^\circ = 4. \]

### Step 2: Simplifying the Denominator

1. **Using Trigonometric Identities**:
For \(\sin 56^\circ \sec 34^\circ\), we know that \(\sec 34^\circ = \frac{1}{\cos 34^\circ}\). Thus:
\[ \sin 56^\circ \sec 34^\circ = \frac{\sin 56^\circ}{\cos 34^\circ}. \]
Also, since \(34^\circ = 90^\circ - 56^\circ\), we have \(\cos 34^\circ = \sin 56^\circ\).

Therefore:
\[ \sin 56^\circ \sec 34^\circ = \frac{\sin 56^\circ}{\sin 56^\circ} = 1. \]

2. **For the Second Part**:
For \(\cos 25^\circ \csc 65^\circ\), we know that \(\csc 65^\circ = \frac{1}{\sin 65^\circ}\) and \(\sin 65^\circ = \cos 25^\circ\) (since \(65^\circ = 90^\circ - 25^\circ\)):
\[ \cos 25^\circ \csc 65^\circ = \frac{\cos 25^\circ}{\cos 25^\circ} = 1. \]

3. **Combining the Parts**:
Therefore, the denominator simplifies to:
\[ \sin 56^\circ \sec 34^\circ + \cos 25^\circ \csc 65^\circ = 1 + 1 = 2. \]

### Step 3: Final Calculation

Now we can substitute the simplified numerator and denominator back into the original expression:
\[ \frac{4}{2} = 2. \]

Thus, the final value of the expression is:

\[ \boxed{2}. \]