If `y=(sinx+cosecx)^2+(cosx+secx)^2` then the minimum value of `y,AAx in R`, is

Options

A. 7
B. 3
C. 9
D. 0

Explanation

To find the minimum value of the expression \( y = (\sin x + \csc x)^2 + (\cos x + \sec x)^2 \), we will follow these steps:

### Step 1: Expand the expression
We start by expanding both squares in the expression:

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

Expanding the first term:

\[ (\sin x + \csc x)^2 = \sin^2 x + 2\sin x \csc x + \csc^2 x \]

Since \( \csc x = \frac{1}{\sin x} \), we have:

\[ 2\sin x \csc x = 2 \]

And \( \csc^2 x = 1 + \cot^2 x \). Therefore,

\[ (\sin x + \csc x)^2 = \sin^2 x + 2 + 1 + \cot^2 x = \sin^2 x + \cot^2 x + 3 \]

Now expanding the second term:

\[ (\cos x + \sec x)^2 = \cos^2 x + 2\cos x \sec x + \sec^2 x \]

Since \( \sec x = \frac{1}{\cos x} \), we have:

\[ 2\cos x \sec x = 2 \]

And \( \sec^2 x = 1 + \tan^2 x \). Therefore,

\[ (\cos x + \sec x)^2 = \cos^2 x + 2 + 1 + \tan^2 x = \cos^2 x + \tan^2 x + 3 \]

### Step 2: Combine the expanded terms
Now we combine both expanded terms:

\[ y = (\sin^2 x + \cot^2 x + 3) + (\cos^2 x + \tan^2 x + 3) \]

This simplifies to:

\[ y = \sin^2 x + \cos^2 x + \cot^2 x + \tan^2 x + 6 \]

Using the identity \( \sin^2 x + \cos^2 x = 1 \):

\[ y = 1 + \cot^2 x + \tan^2 x + 6 = \cot^2 x + \tan^2 x + 7 \]

### Step 3: Simplify \(\cot^2 x + \tan^2 x\)
Recall that:

\[ \tan^2 x = \frac{\sin^2 x}{\cos^2 x} \quad \text{and} \quad \cot^2 x = \frac{\cos^2 x}{\sin^2 x} \]

Let \( t = \tan^2 x \). Then, \( \cot^2 x = \frac{1}{t} \). Thus,

\[ y = t + \frac{1}{t} + 7 \]

### Step 4: Find the minimum value of \( t + \frac{1}{t} \)
The expression \( t + \frac{1}{t} \) has a minimum value of 2 when \( t = 1 \) (by AM-GM inequality).

Thus,

\[ y \geq 2 + 7 = 9 \]

### Conclusion
The minimum value of \( y \) is:

\[ \boxed{9} \]