Find the value of
`(sinx+cosecx)^(2)+(cosx+secx)^(2)-(tan^(2)x+cot^(2)x)`.

Options

A. 8
B. 7
C. 9
D. 10

Answer

To find the value of the expression \((\sin x + \csc x)^2 + (\cos x + \sec x)^2 - (\tan^2 x + \cot^2 x)\), we will simplify it step by step.

### Step 1: Expand the squares
We start by expanding the squares in the expression:
\[ (\sin x + \csc x)^2 = \sin^2 x + 2\sin x \csc x + \csc^2 x \]
\[ (\cos x + \sec x)^2 = \cos^2 x + 2\cos x \sec x + \sec^2 x \]

### Step 2: Substitute the identities
We know that:
\[ \csc x = \frac{1}{\sin x}, \quad \sec x = \frac{1}{\cos x}, \quad \tan x = \frac{\sin x}{\cos x}, \quad \cot x = \frac{\cos x}{\sin x} \]
Substituting these into our expanded squares:
\[ \sin^2 x + 2 + \csc^2 x + \cos^2 x + 2 + \sec^2 x \]

### Step 3: Combine the terms
Now we combine the terms:
\[ \sin^2 x + \cos^2 x + \csc^2 x + \sec^2 x + 4 \]
Using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\):
\[ 1 + \csc^2 x + \sec^2 x + 4 \]

### Step 4: Use the identities for \(\csc^2 x\) and \(\sec^2 x\)
We know:
\[ \csc^2 x = 1 + \cot^2 x \quad \text{and} \quad \sec^2 x = 1 + \tan^2 x \]
Substituting these into the expression:
\[ 1 + (1 + \cot^2 x) + (1 + \tan^2 x) + 4 \]
This simplifies to:
\[ 1 + 1 + 1 + 4 + \tan^2 x + \cot^2 x = 7 + \tan^2 x + \cot^2 x \]

### Step 5: Substitute back into the original expression
Now we substitute back into the original expression:
\[ 7 + \tan^2 x + \cot^2 x - (\tan^2 x + \cot^2 x) \]
The \(\tan^2 x + \cot^2 x\) terms cancel out:
\[ 7 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{7} \]