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: Rewrite the expression
The expression is:
\[ \sin 10^\circ \sec 80^\circ + 4 \tan 45^\circ \]

### Step 2: Evaluate \( \tan 45^\circ \)
We know that:
\[ \tan 45^\circ = 1 \]
So, we can substitute this value into the expression:
\[ \sin 10^\circ \sec 80^\circ + 4 \cdot 1 = \sin 10^\circ \sec 80^\circ + 4 \]

### Step 3: Rewrite \( \sec 80^\circ \)
Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Therefore:
\[ \sec 80^\circ = \frac{1}{\cos 80^\circ} \]
Substituting this into the expression gives:
\[ \sin 10^\circ \cdot \frac{1}{\cos 80^\circ} + 4 \]

### Step 4: Use the complementary angle identity
Using the identity \( \cos(90^\circ - \theta) = \sin \theta \), we have:
\[ \cos 80^\circ = \sin 10^\circ \]
Thus, we can rewrite the expression as:
\[ \sin 10^\circ \cdot \frac{1}{\sin 10^\circ} + 4 \]

### Step 5: Simplify the expression
Now, simplifying \( \sin 10^\circ \cdot \frac{1}{\sin 10^\circ} \):
\[ 1 + 4 \]

### Step 6: Final calculation
Now, adding the terms together:
\[ 1 + 4 = 5 \]

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