Express the following in terms of angles between `0^(@)` and `45^(@)`
`sin 59^(@)+tan 63^(@)`

Answer

To express \( \sin 59^\circ + \tan 63^\circ \) in terms of angles between \( 0^\circ \) and \( 45^\circ \), we can use the complementary angle identities for sine and tangent.

### Step-by-Step Solution:

1. **Identify the complementary angle identities**:
- We know that:
\[ \sin(90^\circ - \theta) = \cos(\theta) \]
\[ \tan(90^\circ - \theta) = \cot(\theta) \]

2. **Rewrite \( \sin 59^\circ \)**:
- Using the identity for sine, we can express \( \sin 59^\circ \) as:
\[ \sin 59^\circ = \cos(90^\circ - 59^\circ) = \cos 31^\circ \]

3. **Rewrite \( \tan 63^\circ \)**:
- Using the identity for tangent, we can express \( \tan 63^\circ \) as:
\[ \tan 63^\circ = \cot(90^\circ - 63^\circ) = \cot 27^\circ \]

4. **Combine the expressions**:
- Now we can rewrite the original expression:
\[ \sin 59^\circ + \tan 63^\circ = \cos 31^\circ + \cot 27^\circ \]

5. **Check the angle ranges**:
- Both \( 31^\circ \) and \( 27^\circ \) are between \( 0^\circ \) and \( 45^\circ \), so we have successfully expressed the original expression in terms of angles within the desired range.

### Final Answer:
\[ \sin 59^\circ + \tan 63^\circ = \cos 31^\circ + \cot 27^\circ \]