Evaluate
` cos 40 ^(@) cosec 50 ^(@)+ sin 50 ^(@) sec 40 ^(@) `

Answer

To evaluate the expression \( \cos 40^\circ \csc 50^\circ + \sin 50^\circ \sec 40^\circ \), we will follow these steps:

### Step 1: Rewrite \(\csc 50^\circ\) and \(\sec 40^\circ\)
We know that:
\[ \csc 50^\circ = \frac{1}{\sin 50^\circ} \]
and
\[ \sec 40^\circ = \frac{1}{\cos 40^\circ} \]

### Step 2: Substitute the identities into the expression
Substituting these identities into the original expression gives us:
\[ \cos 40^\circ \cdot \frac{1}{\sin 50^\circ} + \sin 50^\circ \cdot \frac{1}{\cos 40^\circ} \]
This simplifies to:
\[ \frac{\cos 40^\circ}{\sin 50^\circ} + \frac{\sin 50^\circ}{\cos 40^\circ} \]

### Step 3: Use the identity \(\sin 50^\circ = \cos(90^\circ - 50^\circ) = \cos 40^\circ\)
Since \(\sin 50^\circ = \cos 40^\circ\), we can rewrite the expression:
\[ \frac{\cos 40^\circ}{\cos 40^\circ} + \frac{\cos 40^\circ}{\cos 40^\circ} \]
This simplifies to:
\[ 1 + 1 \]

### Step 4: Final Calculation
Adding these values together gives:
\[ 1 + 1 = 2 \]

### Conclusion
Thus, the value of the expression \( \cos 40^\circ \csc 50^\circ + \sin 50^\circ \sec 40^\circ \) is:
\[ \boxed{2} \]