Find the value of
`(sinx+cosecx)^(2)+(cos+secx)^(2)-(tanx+cotx)^(2)`,

Options

A. 7
B. 5
C. 9
D. 4

Answer

To solve the expression \((\sin x + \csc x)^2 + (\cos x + \sec x)^2 - (\tan x + \cot x)^2\), we will break it down step by step.

### Step 1: Expand \((\sin x + \csc x)^2\)
Using the identity \((a + b)^2 = a^2 + b^2 + 2ab\):
\[ (\sin x + \csc x)^2 = \sin^2 x + \csc^2 x + 2 \sin x \csc x \]
Since \(\csc x = \frac{1}{\sin x}\), we have:
\[ 2 \sin x \csc x = 2 \sin x \cdot \frac{1}{\sin x} = 2 \]
Thus,
\[ (\sin x + \csc x)^2 = \sin^2 x + \csc^2 x + 2 \]

### Step 2: Expand \((\cos x + \sec x)^2\)
Similarly, we expand:
\[ (\cos x + \sec x)^2 = \cos^2 x + \sec^2 x + 2 \cos x \sec x \]
Since \(\sec x = \frac{1}{\cos x}\), we have:
\[ 2 \cos x \sec x = 2 \cos x \cdot \frac{1}{\cos x} = 2 \]
Thus,
\[ (\cos x + \sec x)^2 = \cos^2 x + \sec^2 x + 2 \]

### Step 3: Expand \((\tan x + \cot x)^2\)
Now we expand:
\[ (\tan x + \cot x)^2 = \tan^2 x + \cot^2 x + 2 \tan x \cot x \]
Since \(\tan x = \frac{\sin x}{\cos x}\) and \(\cot x = \frac{\cos x}{\sin x}\), we have:
\[ 2 \tan x \cot x = 2 \cdot \frac{\sin x}{\cos x} \cdot \frac{\cos x}{\sin x} = 2 \]
Thus,
\[ (\tan x + \cot x)^2 = \tan^2 x + \cot^2 x + 2 \]

### Step 4: Substitute back into the original expression
Now we substitute back into the original expression:
\[ (\sin^2 x + \csc^2 x + 2) + (\cos^2 x + \sec^2 x + 2) - (\tan^2 x + \cot^2 x + 2) \]
This simplifies to:
\[ \sin^2 x + \csc^2 x + \cos^2 x + \sec^2 x + 4 - (\tan^2 x + \cot^2 x + 2) \]

### Step 5: Simplify further
Now we simplify:
\[ \sin^2 x + \cos^2 x = 1 \]
And we know:
\[ \csc^2 x = 1 + \cot^2 x \quad \text{and} \quad \sec^2 x = 1 + \tan^2 x \]
Thus,
\[ \csc^2 x + \sec^2 x = (1 + \cot^2 x) + (1 + \tan^2 x) = 2 + \tan^2 x + \cot^2 x \]
Now substituting this back:
\[ 1 + (2 + \tan^2 x + \cot^2 x) + 4 - (\tan^2 x + \cot^2 x + 2) \]
This simplifies to:
\[ 1 + 2 + 4 + 2 - 2 = 7 - 2 = 5 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{5} \]