The equation `cosx+sinx=2` has

Options

A. Only one solution
B. Two solution
C. No solution
D. Infinite number of solutions

Answer

To solve the equation \( \cos x + \sin x = 2 \) and determine how many solutions it has, we can follow these steps:

### Step 1: Understand the Range of Sine and Cosine Functions
The sine and cosine functions have specific ranges:
- The maximum value of \( \cos x \) is 1.
- The maximum value of \( \sin x \) is also 1.

### Step 2: Analyze the Equation
The equation given is:
\[ \cos x + \sin x = 2 \]
Since both \( \cos x \) and \( \sin x \) can each only reach a maximum of 1, the maximum value of \( \cos x + \sin x \) is:
\[ 1 + 1 = 2 \]
This means that the left-hand side of the equation can only equal 2 if both \( \cos x \) and \( \sin x \) are at their maximum values simultaneously.

### Step 3: Determine Conditions for Maximum Values
For \( \cos x + \sin x \) to equal 2, we need:
\[ \cos x = 1 \quad \text{and} \quad \sin x = 1 \]
However, it is impossible for both \( \cos x \) and \( \sin x \) to be 1 at the same time. The cosine function reaches its maximum value of 1 at \( x = 0 \) (and multiples of \( 2\pi \)), while the sine function reaches its maximum value of 1 at \( x = \frac{\pi}{2} \) (and odd multiples of \( \frac{\pi}{2} \)).

### Step 4: Conclusion
Since there is no value of \( x \) that satisfies both conditions simultaneously, we conclude that:
\[ \cos x + \sin x = 2 \text{ has no solution.} \]

### Final Answer
The equation \( \cos x + \sin x = 2 \) has **no solution**.

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