If `P=cos(cos x)+sin (cos x)`, then the least and greatest value of P respectively.

Options

A. `-1` and 1
B. 0 and 2
C. `-sqrt(2)` and `sqrt(2)`
D. 0 and `sqrt(2)`

Answer

To find the least and greatest values of \( P = \cos(\cos x) + \sin(\cos x) \), we can follow these steps:

### Step 1: Substitute \( \cos x \)
Let \( t = \cos x \). The range of \( t \) is from -1 to 1, since \( \cos x \) varies between -1 and 1 for all \( x \). Thus, we can rewrite \( P \) as:
\[ P = \cos(t) + \sin(t) \]

### Step 2: Find the maximum and minimum values of \( P \)
To find the maximum and minimum values of \( P = \cos(t) + \sin(t) \), we can use the property of trigonometric functions. The maximum and minimum values of \( a \cos(t) + b \sin(t) \) can be calculated using the formula:
\[ \text{Maximum value} = \sqrt{a^2 + b^2} \]
\[ \text{Minimum value} = -\sqrt{a^2 + b^2} \]
In our case, \( a = 1 \) and \( b = 1 \).

### Step 3: Calculate the maximum and minimum values
Now we can calculate:
\[ \text{Maximum value} = \sqrt{1^2 + 1^2} = \sqrt{2} \]
\[ \text{Minimum value} = -\sqrt{1^2 + 1^2} = -\sqrt{2} \]

### Step 4: Conclusion
Thus, the least and greatest values of \( P \) are:
- Least value: \( -\sqrt{2} \)
- Greatest value: \( \sqrt{2} \)

### Final Answer
The least and greatest values of \( P \) are \( -\sqrt{2} \) and \( \sqrt{2} \) respectively.

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