Maximum value of `(cos theta-sin theta)` is :

Options

A. `sqrt2`
B. 1
C. `(1)/(2)`
D. `(1)/(sqrt2)`

Answer

To find the maximum value of \( \cos \theta - \sin \theta \), we can use the formula for the maximum value of the expression \( a \cos \theta + b \sin \theta \).

1. **Identify the coefficients**: In our case, we have:
- \( a = 1 \) (coefficient of \( \cos \theta \))
- \( b = -1 \) (coefficient of \( -\sin \theta \))

2. **Use the maximum value formula**: The maximum value of \( a \cos \theta + b \sin \theta \) is given by:
\[ \sqrt{a^2 + b^2} \]

3. **Substitute the values**: Plugging in our values for \( a \) and \( b \):
\[ \sqrt{1^2 + (-1)^2} \]

4. **Calculate the squares**: This simplifies to:
\[ \sqrt{1 + 1} = \sqrt{2} \]

5. **Conclusion**: Therefore, the maximum value of \( \cos \theta - \sin \theta \) is \( \sqrt{2} \).