If `sinx+siny+sinz+sinw=-4" then the value of "sin^400x+sin^300y+sin^200z+sin^100w` is

Options

A. `sin^400x.sin^300y.sin^200z.sin^100w`
B. `sinx.siny.sinz.sinw`
C. 4
D. 3

Explanation

To solve the problem, we need to find the value of \( \sin^{400}x + \sin^{300}y + \sin^{200}z + \sin^{100}w \) given that \( \sin x + \sin y + \sin z + \sin w = -4 \).

### Step-by-step Solution:

1. **Understanding the Range of Sine Function**:
The sine function, \( \sin \theta \), has a range of values between -1 and 1. Therefore, the minimum value of \( \sin x + \sin y + \sin z + \sin w \) occurs when each sine term is at its minimum value, which is -1.

2. **Setting Up the Equation**:
Given that \( \sin x + \sin y + \sin z + \sin w = -4 \), the only way this equation can hold true is if each sine term equals -1. Thus, we have:
\[ \sin x = \sin y = \sin z = \sin w = -1 \]

3. **Calculating the Powers of Sine Values**:
Now, we need to compute \( \sin^{400}x + \sin^{300}y + \sin^{200}z + \sin^{100}w \):
\[ \sin^{400}x = (-1)^{400} = 1 \]
\[ \sin^{300}y = (-1)^{300} = 1 \]
\[ \sin^{200}z = (-1)^{200} = 1 \]
\[ \sin^{100}w = (-1)^{100} = 1 \]

4. **Adding the Results**:
Now, we can add these values together:
\[ \sin^{400}x + \sin^{300}y + \sin^{200}z + \sin^{100}w = 1 + 1 + 1 + 1 = 4 \]

5. **Final Answer**:
Therefore, the value of \( \sin^{400}x + \sin^{300}y + \sin^{200}z + \sin^{100}w \) is \( \boxed{4} \).