The minimum value of
`|sinx+cosx+tanx+secx+"cosec"x+cotx|`, is

Options

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

Answer

To find the minimum value of the expression \( | \sin x + \cos x + \tan x + \sec x + \csc x + \cot x | \), we can follow these steps:

### Step 1: Rewrite the expression
Let:
\[ I = \sin x + \cos x + \tan x + \sec x + \csc x + \cot x \]
We can express \( \tan x \), \( \sec x \), \( \csc x \), and \( \cot x \) in terms of \( \sin x \) and \( \cos x \):
\[ I = \sin x + \cos x + \frac{\sin x}{\cos x} + \frac{1}{\cos x} + \frac{1}{\sin x} + \frac{\cos x}{\sin x} \]

### Step 2: Combine terms
Now, we can combine the terms:
\[ I = \sin x + \cos x + \frac{\sin^2 x + 1 + \cos^2 x}{\sin x \cos x} \]
Using the identity \( \sin^2 x + \cos^2 x = 1 \):
\[ I = \sin x + \cos x + \frac{1 + 1}{\sin x \cos x} = \sin x + \cos x + \frac{2}{\sin x \cos x} \]

### Step 3: Substitute \( \sin x + \cos x \)
Let \( \alpha = \sin x + \cos x \). We know:
\[ \alpha^2 = \sin^2 x + \cos^2 x + 2\sin x \cos x = 1 + 2\sin x \cos x \]
Thus, we can express \( \sin x \cos x \) in terms of \( \alpha \):
\[ \sin x \cos x = \frac{\alpha^2 - 1}{2} \]

### Step 4: Substitute back into \( I \)
Now substituting back into \( I \):
\[ I = \alpha + \frac{2}{\frac{\alpha^2 - 1}{2}} = \alpha + \frac{4}{\alpha^2 - 1} \]

### Step 5: Find the minimum value of \( I \)
To find the minimum value of \( I \), we need to differentiate it with respect to \( \alpha \) and set the derivative to zero. However, we can also use the AM-GM inequality:
\[ \alpha + \frac{4}{\alpha^2 - 1} \geq 2\sqrt{\alpha \cdot \frac{4}{\alpha^2 - 1}} \]
This gives us a lower bound for \( I \).

### Step 6: Analyze the bounds
Using the properties of \( \alpha \):
\[ - \sqrt{2} \leq \alpha \leq \sqrt{2} \]
The minimum occurs when \( \alpha \) is at its maximum or minimum values.

### Step 7: Evaluate at critical points
Evaluating \( I \) at \( \alpha = \sqrt{2} \):
\[ I = \sqrt{2} + \frac{4}{(\sqrt{2})^2 - 1} = \sqrt{2} + \frac{4}{2 - 1} = \sqrt{2} + 4 \]
Evaluating \( I \) at \( \alpha = -\sqrt{2} \):
\[ I = -\sqrt{2} + \frac{4}{(-\sqrt{2})^2 - 1} = -\sqrt{2} + \frac{4}{2 - 1} = -\sqrt{2} + 4 \]

### Step 8: Find the minimum value
The minimum value of \( |I| \) occurs at:
\[ |I| = |-\sqrt{2} + 4| = 4 - \sqrt{2} \]

### Final Result
Thus, the minimum value of \( | \sin x + \cos x + \tan x + \sec x + \csc x + \cot x | \) is:
\[ \boxed{2\sqrt{2} - 1} \]