Express `cos70^(@)+sin70^(@)+tan65^(@)` in terms of trigonometric ratios of angles lying between `0^(@)and45^(@)`

Answer

To express \( \cos 70^\circ + \sin 70^\circ + \tan 65^\circ \) in terms of trigonometric ratios of angles lying between \( 0^\circ \) and \( 45^\circ \), we can use the complementary angle identities.

### Step-by-Step Solution:

1. **Rewrite \( \cos 70^\circ \)**:
\[ \cos 70^\circ = \sin(90^\circ - 70^\circ) = \sin 20^\circ \]

**Hint**: Use the identity \( \cos(90^\circ - \theta) = \sin \theta \).

2. **Rewrite \( \sin 70^\circ \)**:
\[ \sin 70^\circ = \cos(90^\circ - 70^\circ) = \cos 20^\circ \]

**Hint**: Use the identity \( \sin(90^\circ - \theta) = \cos \theta \).

3. **Rewrite \( \tan 65^\circ \)**:
\[ \tan 65^\circ = \cot(90^\circ - 65^\circ) = \cot 25^\circ \]

**Hint**: Use the identity \( \tan(90^\circ - \theta) = \cot \theta \).

4. **Combine the expressions**:
Now, substituting these values back into the original expression:
\[ \cos 70^\circ + \sin 70^\circ + \tan 65^\circ = \sin 20^\circ + \cos 20^\circ + \cot 25^\circ \]

**Hint**: Combine all the rewritten terms to form the final expression.

### Final Answer:
\[ \sin 20^\circ + \cos 20^\circ + \cot 25^\circ \]