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(cosec^2x-1)` whenever it is defined is

Options

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

Explanation

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 step by step.

### Step 1: Simplify each term
1. The first term:
\[ \frac{\sin x}{\sqrt{1 - \cos^2 x}} = \frac{\sin x}{\sqrt{\sin^2 x}} = \frac{\sin x}{|\sin x|}. \]
This equals 1 if \(\sin x > 0\) (first and second quadrants) and -1 if \(\sin x < 0\) (third and fourth quadrants).

2. The second term:
\[ \frac{\cos x}{\sqrt{1 - \sin^2 x}} = \frac{\cos x}{\sqrt{\cos^2 x}} = \frac{\cos x}{|\cos x|}. \]
This equals 1 if \(\cos x > 0\) (first and fourth quadrants) and -1 if \(\cos x < 0\) (second and third quadrants).

3. The third term:
\[ \frac{\tan x}{\sqrt{\sec^2 x - 1}} = \frac{\tan x}{\sqrt{\tan^2 x}} = \frac{\tan x}{|\tan x|}. \]
This equals 1 if \(\tan x > 0\) (first and third quadrants) and -1 if \(\tan x < 0\) (second and fourth quadrants).

4. The fourth term:
\[ \frac{\cot x}{\sqrt{\csc^2 x - 1}} = \frac{\cot x}{\sqrt{\cot^2 x}} = \frac{\cot x}{|\cot x|}. \]
This equals 1 if \(\cot x > 0\) (first and fourth quadrants) and -1 if \(\cot x < 0\) (second and third quadrants).

### Step 2: Analyze the function in different quadrants
Now we will analyze \(f(x)\) in each quadrant:

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

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

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

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

### Step 3: Determine the minimum value
From the analysis:
- In the first quadrant, \(f(x) = 4\).
- In the second quadrant, \(f(x) = -2\).
- In the third quadrant, \(f(x) = 0\).
- In the fourth quadrant, \(f(x) = 0\).

The minimum value of \(f(x)\) occurs in the second quadrant, where \(f(x) = -2\).

Thus, the minimum value of the function is

\[ \boxed{-2}. \]