Find the value of ` sin 48^(@) sec 42^(@)+cos48^(@) "cosec"42^(@).`

Explanation

To solve the expression \( \sin 48^\circ \sec 42^\circ + \cos 48^\circ \csc 42^\circ \), we can follow these steps:

### Step 1: Rewrite the secant and cosecant functions
We know that:
\[ \sec \theta = \frac{1}{\cos \theta} \quad \text{and} \quad \csc \theta = \frac{1}{\sin \theta} \]
Thus, we can rewrite the expression:
\[ \sin 48^\circ \sec 42^\circ + \cos 48^\circ \csc 42^\circ = \sin 48^\circ \cdot \frac{1}{\cos 42^\circ} + \cos 48^\circ \cdot \frac{1}{\sin 42^\circ} \]

### Step 2: Combine the terms
Now, we can combine the terms over a common denominator:
\[ = \frac{\sin 48^\circ}{\cos 42^\circ} + \frac{\cos 48^\circ}{\sin 42^\circ} \]
The common denominator is \( \cos 42^\circ \sin 42^\circ \):
\[ = \frac{\sin 48^\circ \sin 42^\circ + \cos 48^\circ \cos 42^\circ}{\cos 42^\circ \sin 42^\circ} \]

### Step 3: Use the cosine addition formula
We can use the cosine addition formula:
\[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
Here, let \( A = 48^\circ \) and \( B = 42^\circ \):
\[ \sin 48^\circ \sin 42^\circ + \cos 48^\circ \cos 42^\circ = \cos(48^\circ - 42^\circ) = \cos 6^\circ \]

### Step 4: Substitute back into the expression
Now substitute this back into our expression:
\[ = \frac{\cos 6^\circ}{\cos 42^\circ \sin 42^\circ} \]

### Step 5: Simplify further
We can leave the expression as is or evaluate \( \cos 6^\circ \) if needed. However, it is often sufficient to leave it in this form unless a numerical value is required.

### Final Answer
Thus, the final value of the expression \( \sin 48^\circ \sec 42^\circ + \cos 48^\circ \csc 42^\circ \) is:
\[ \frac{\cos 6^\circ}{\cos 42^\circ \sin 42^\circ} \]