Evaluate `sin 10^(@)sec80^(@)+4tan45^(@)`.

Answer

To evaluate the expression \( \sin 10^\circ \sec 80^\circ + 4 \tan 45^\circ \), we can follow these steps:

### Step 1: Understand the components of the expression
We have two parts in the expression: \( \sin 10^\circ \sec 80^\circ \) and \( 4 \tan 45^\circ \).

### Step 2: Simplify \( \tan 45^\circ \)
We know that:
\[ \tan 45^\circ = 1 \]
Thus, we can simplify the second part of the expression:
\[ 4 \tan 45^\circ = 4 \times 1 = 4 \]

### Step 3: Rewrite \( \sec 80^\circ \)
We can use the identity \( \sec \theta = \frac{1}{\cos \theta} \) to rewrite \( \sec 80^\circ \):
\[ \sec 80^\circ = \frac{1}{\cos 80^\circ} \]

### Step 4: Use the co-function identity
We know from trigonometric identities that:
\[ \cos 80^\circ = \sin (90^\circ - 80^\circ) = \sin 10^\circ \]
Thus, we can rewrite \( \sec 80^\circ \):
\[ \sec 80^\circ = \frac{1}{\sin 10^\circ} \]

### Step 5: Substitute back into the expression
Now we can substitute \( \sec 80^\circ \) back into the expression:
\[ \sin 10^\circ \sec 80^\circ = \sin 10^\circ \cdot \frac{1}{\sin 10^\circ} = 1 \]

### Step 6: Combine the results
Now we can combine the results from Step 2 and Step 5:
\[ \sin 10^\circ \sec 80^\circ + 4 = 1 + 4 = 5 \]

### Final Answer
Thus, the value of the expression \( \sin 10^\circ \sec 80^\circ + 4 \tan 45^\circ \) is:
\[ \boxed{5} \]