If `lambda` be the minimum value of `y= (sinx+"cosec"x)^(2)+(cosx+secx)^(2)+(tanx+cot x)^(2)` where `x in R`. Find `lambda-6`.

Answer

To find the minimum value of the expression

\[ y = (\sin x + \csc x)^2 + (\cos x + \sec x)^2 + (\tan x + \cot x)^2 \]

we will proceed step by step.

### Step 1: Expand Each Term
We start by expanding each term in the expression.

1. **Expand \((\sin x + \csc x)^2\)**:
\[ (\sin x + \csc x)^2 = \sin^2 x + 2 + \csc^2 x \]
where \(\csc x = \frac{1}{\sin x}\), thus \(\csc^2 x = \frac{1}{\sin^2 x}\).

2. **Expand \((\cos x + \sec x)^2\)**:
\[ (\cos x + \sec x)^2 = \cos^2 x + 2 + \sec^2 x \]
where \(\sec x = \frac{1}{\cos x}\), thus \(\sec^2 x = \frac{1}{\cos^2 x}\).

3. **Expand \((\tan x + \cot x)^2\)**:
\[ (\tan x + \cot x)^2 = \tan^2 x + 2 + \cot^2 x \]
where \(\tan x = \frac{\sin x}{\cos x}\) and \(\cot x = \frac{\cos x}{\sin x}\).

### Step 2: Combine the Expanded Terms
Now we can combine all the expanded terms:
\[ y = \left(\sin^2 x + \csc^2 x + 2\right) + \left(\cos^2 x + \sec^2 x + 2\right) + \left(\tan^2 x + \cot^2 x + 2\right) \]

Combining these gives:
\[ y = \sin^2 x + \cos^2 x + \csc^2 x + \sec^2 x + \tan^2 x + \cot^2 x + 6 \]

### Step 3: Use Trigonometric Identities
Using the identities:
- \(\sin^2 x + \cos^2 x = 1\)
- \(\csc^2 x = 1 + \cot^2 x\)
- \(\sec^2 x = 1 + \tan^2 x\)

We can substitute:
\[ y = 1 + (1 + \cot^2 x) + (1 + \tan^2 x) + \tan^2 x + \cot^2 x + 6 \]
This simplifies to:
\[ y = 9 + \tan^2 x + \cot^2 x + \tan^2 x + \cot^2 x \]
\[ y = 9 + 2(\tan^2 x + \cot^2 x) \]

### Step 4: Simplify Further
Using the identity \(\tan^2 x + \cot^2 x \geq 2\) (AM-GM inequality), we find:
\[ \tan^2 x + \cot^2 x \geq 2 \]
Thus:
\[ y \geq 9 + 2 \cdot 2 = 13 \]

### Step 5: Find Minimum Value
The minimum value of \(y\) is \(13\).

### Step 6: Calculate \(\lambda - 6\)
Since \(\lambda\) is the minimum value of \(y\):
\[ \lambda = 13 \]
Thus:
\[ \lambda - 6 = 13 - 6 = 7 \]

### Final Answer
\[ \lambda - 6 = 7 \]