If `90^(@) lt alpha lt 180^(@) and 270^(@) lt beta lt 360^(@)`, then which of the following cannot be true?

Options

A. `sinalpha=sinbeta`
B. `tanalpha=sinbeta`
C. `tanalpha=tanbeta`
D. `sinalpha=cosbeta`

Explanation

To solve the problem, we need to analyze the given ranges of angles \( \alpha \) and \( \beta \) and evaluate the truth of the statements provided in the options.

### Step-by-Step Solution:

1. **Identify the Ranges of Angles**:
- We have \( 90^\circ < \alpha < 180^\circ \), which means \( \alpha \) is in the second quadrant.
- We have \( 270^\circ < \beta < 360^\circ \), which means \( \beta \) is in the fourth quadrant.

2. **Understand the Sine Function in Quadrants**:
- In the second quadrant (where \( \alpha \) lies), the sine function is positive. Therefore, \( \sin(\alpha) > 0 \).
- In the fourth quadrant (where \( \beta \) lies), the sine function is negative. Therefore, \( \sin(\beta) < 0 \).

3. **Evaluate the Statement \( \sin(\alpha) = \sin(\beta) \)**:
- Since \( \sin(\alpha) > 0 \) and \( \sin(\beta) < 0 \), it is impossible for \( \sin(\alpha) \) to equal \( \sin(\beta) \). Thus, this statement cannot be true.

4. **Conclusion**:
- The statement \( \sin(\alpha) = \sin(\beta) \) is the one that cannot be true given the ranges of \( \alpha \) and \( \beta \).

### Final Answer:
The option that cannot be true is \( \sin(\alpha) = \sin(\beta) \).