The minimum value of the function `f(x)=(sinx)/(sqrt(1-cos^2x))+(cosx)/(sqrt(1-sin^2x))+(tanx)/(sqrt(sec^2x-1))+(cotx)/(sqrt(c o s e c^2x-1))`whenever it is defined is

Options

A. 4
B. -2
C. 0
D. 2

Answer

To find the minimum value of the function

\[ f(x) = \frac{\sin x}{\sqrt{1 - \cos^2 x}} + \frac{\cos x}{\sqrt{1 - \sin^2 x}} + \frac{\tan x}{\sqrt{\sec^2 x - 1}} + \frac{\cot x}{\sqrt{\csc^2 x - 1}}, \]

we can simplify each term using trigonometric identities.

### Step 1: Simplify the Function
Using the identity \( \sin^2 x + \cos^2 x = 1 \), we can rewrite the square roots:
- \( \sqrt{1 - \cos^2 x} = \sqrt{\sin^2 x} = |\sin x| \)
- \( \sqrt{1 - \sin^2 x} = \sqrt{\cos^2 x} = |\cos x| \)
- \( \sqrt{\sec^2 x - 1} = \sqrt{\tan^2 x} = |\tan x| \)
- \( \sqrt{\csc^2 x - 1} = \sqrt{\cot^2 x} = |\cot x| \)

Thus, we can rewrite \( f(x) \) as:

\[ f(x) = \frac{\sin x}{|\sin x|} + \frac{\cos x}{|\cos x|} + \frac{\tan x}{|\tan x|} + \frac{\cot x}{|\cot x|}. \]

### Step 2: Analyze Each Quadrant
Now we will analyze the function in different quadrants since the signs of the trigonometric functions change:

1. **First Quadrant (0 to π/2)**:
- All functions are positive.
- \( f(x) = 1 + 1 + 1 + 1 = 4 \)

2. **Second Quadrant (π/2 to π)**:
- \( \sin x > 0, \cos x < 0 \)
- \( f(x) = 1 - 1 - 1 - 1 = -2 \)

3. **Third Quadrant (π to 3π/2)**:
- \( \sin x < 0, \cos x < 0 \)
- \( f(x) = -1 - 1 + 1 + 1 = 0 \)

4. **Fourth Quadrant (3π/2 to 2π)**:
- \( \sin x < 0, \cos x > 0 \)
- \( f(x) = -1 + 1 - 1 - 1 = -2 \)

### Step 3: Determine the Minimum Value
From the values calculated in each quadrant:
- First Quadrant: \( f(x) = 4 \)
- Second Quadrant: \( f(x) = -2 \)
- Third Quadrant: \( f(x) = 0 \)
- Fourth Quadrant: \( f(x) = -2 \)

The minimum value of \( f(x) \) is \( -2 \).

### Final Answer
Thus, the minimum value of the function \( f(x) \) whenever it is defined is:

\[ \boxed{-2} \]