Evaluate :
`3 cos 80^(@)cosec 10^(@)+2sin 59^(@)sec 31^(@)`

Answer

To evaluate the expression \( 3 \cos 80^\circ \csc 10^\circ + 2 \sin 59^\circ \sec 31^\circ \), we can follow these steps:

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

### Step 2: Rewrite \( \cos 80^\circ \) and \( \sin 59^\circ \)
Using the above identities:
- \( \cos 80^\circ = \sin(90^\circ - 80^\circ) = \sin 10^\circ \)
- \( \sin 59^\circ = \cos(90^\circ - 59^\circ) = \cos 31^\circ \)

### Step 3: Substitute in the Expression
Now substitute these values back into the expression:
\[ 3 \cos 80^\circ \csc 10^\circ + 2 \sin 59^\circ \sec 31^\circ = 3 \sin 10^\circ \csc 10^\circ + 2 \cos 31^\circ \sec 31^\circ \]

### Step 4: Simplify Using Definitions
Recall that:
- \( \csc \theta = \frac{1}{\sin \theta} \)
- \( \sec \theta = \frac{1}{\cos \theta} \)

Thus:
- \( \sin 10^\circ \csc 10^\circ = \sin 10^\circ \cdot \frac{1}{\sin 10^\circ} = 1 \)
- \( \cos 31^\circ \sec 31^\circ = \cos 31^\circ \cdot \frac{1}{\cos 31^\circ} = 1 \)

### Step 5: Substitute Back into the Expression
Now substitute these results back into the expression:
\[ 3 \cdot 1 + 2 \cdot 1 = 3 + 2 = 5 \]

### Final Answer
Thus, the evaluated result is:
\[ \boxed{5} \]