A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.)
1. **Define Sequences**: We define the sequences \( M_k \) and \( m_k \) as the maximum and minimum values achievable with \( k \) presses of the buttons \( \cos \) or \( \sin \).

2. **Initial Values**: Initially, the calculator displays 1, so \( M_0 = m_0 = 1 \).

3. **Monotonic Properties**: Since \( \sin \) and \( \cos \) are monotonic functions in the first quadrant (between 0 and \( \pi/2 \)), we can use the following properties:
   - \( \sin(x) > \cos(y) \Leftrightarrow x + y > \pi/2 \)
   - \( \sin(x) < \cos(y) \Leftrightarrow x + y < \pi/2 \)
   - \( \sin(x) > \sin(y) \Leftrightarrow x > y \)
   - \( \cos(x) > \cos(y) \Leftrightarrow x < y \)

4. **Update Rules**: Based on these properties, we can update \( M_k \) and \( m_k \) as follows:
   - If \( M_k + m_k > \pi/2 \), then \( M_{k+1} = \sin(M_k) \) and \( m_{k+1} = \cos(M_k) \).
   - If \( M_k + m_k < \pi/2 \), then \( M_{k+1} = \cos(m_k) \) and \( m_{k+1} = \sin(m_k) \).

5. **Invariant**: We observe that \( M_k^2 + m_k^2 = 1 \) for \( k \geq 1 \). This is because \( \sin^2(x) + \cos^2(x) = 1 \).

6. **Power Mean Inequality**: Using the Power Mean Inequality, we can assert that \( M_k + m_k \leq \sqrt{2} \leq \pi/2 \). Therefore, we can use the update rule \( M_{k+1} = \cos(m_k) \) and \( m_{k+1} = \sin(m_k) \) for any \( k \geq 1 \).

7. **Iteration**: By iterating the update rules, we can determine the maximum value achievable after 2001 presses. Since \( \cos \) and \( \sin \) are bounded by 1, the maximum value achievable is 1.

The final answer is \(\boxed{1}\).