If `alpha,beta` is obtuse angle & `alphagtbeta`.
`(90^(@)lt(alpha&beta)lt180^(@))` then which options are correct?

Options

A. `sinalphagtsinbeta`
B. `sinalphaltsinbeta`
C. `cosalphagtcosbeta`
D. `cosalphaltcosbeta`

Answer

To solve the problem, we need to analyze the properties of the sine and cosine functions for obtuse angles (angles between 90° and 180°). Given that α and β are obtuse angles and α > β, we can derive the relationships between their sine and cosine values.

### Step-by-Step Solution:

1. **Understanding the Range of Angles**:
- Since α and β are obtuse angles, we know that:
\[ 90^\circ < \alpha < 180^\circ \quad \text{and} \quad 90^\circ < \beta < 180^\circ \]

2. **Properties of Sine and Cosine in the Range**:
- The sine function (sin) is increasing from 90° to 90° and then starts decreasing from 90° to 180°.
- The cosine function (cos) is decreasing from 90° to 180°.

3. **Comparing Sine Values**:
- Since α > β and both angles are in the range (90°, 180°):
- The sine function is decreasing in this range, which means:
\[ \sin(\alpha) < \sin(\beta) \]
- Therefore, the first option "sin α > sin β" is **incorrect**, and the second option "sin α < sin β" is **correct**.

4. **Comparing Cosine Values**:
- Similarly, for the cosine function:
- Since α > β and both angles are in the range (90°, 180°):
\[ \cos(\alpha) < \cos(\beta) \]
- Thus, the third option "cos α > cos β" is **incorrect**, and the fourth option "cos α < cos β" is **correct**.

5. **Conclusion**:
- The correct options are:
- **sin α < sin β** (Option 2)
- **cos α < cos β** (Option 4)

### Final Answer:
The correct options are:
- sin α < sin β (Option 2)
- cos α < cos β (Option 4)