Find the value of `(sec 65^@)/(cosec 25^@)`

Answer

To find the value of \(\frac{\sec 65^\circ}{\csc 25^\circ}\), we can follow these steps:

### Step 1: Rewrite the Cosecant Function
We know that \(\csc \theta = \frac{1}{\sin \theta}\). Therefore, we can rewrite \(\csc 25^\circ\) as:
\[ \csc 25^\circ = \frac{1}{\sin 25^\circ} \]
Thus, the expression becomes:
\[ \frac{\sec 65^\circ}{\csc 25^\circ} = \sec 65^\circ \cdot \sin 25^\circ \]

### Step 2: Use the Co-Function Identity
Using the co-function identity, we know that:
\[ \sin(90^\circ - \theta) = \cos \theta \]
So, we can express \(\sin 25^\circ\) as:
\[ \sin 25^\circ = \cos(90^\circ - 25^\circ) = \cos 65^\circ \]
Now we can substitute this back into our expression:
\[ \frac{\sec 65^\circ}{\csc 25^\circ} = \sec 65^\circ \cdot \cos 65^\circ \]

### Step 3: Rewrite the Secant Function
We know that \(\sec \theta = \frac{1}{\cos \theta}\). Therefore:
\[ \sec 65^\circ = \frac{1}{\cos 65^\circ} \]
Substituting this into our expression gives:
\[ \frac{\sec 65^\circ}{\csc 25^\circ} = \frac{1}{\cos 65^\circ} \cdot \cos 65^\circ \]

### Step 4: Simplify the Expression
Now, we can simplify:
\[ \frac{1}{\cos 65^\circ} \cdot \cos 65^\circ = 1 \]

### Final Answer
Thus, the value of \(\frac{\sec 65^\circ}{\csc 25^\circ}\) is:
\[ \boxed{1} \]