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

Options

A. `-7`
B. 5
C. 1
D. 4

Answer

To solve the expression \( ( \sin x + \csc x )^2 + ( \cos x - \sec x )^2 - ( \tan x + \cot x )^2 \), we can follow these steps:

### Step 1: Expand the squares
Using the algebraic identity \( (a + b)^2 = a^2 + b^2 + 2ab \) and \( (a - b)^2 = a^2 + b^2 - 2ab \), we can expand each term.

1. **Expand \( ( \sin x + \csc x )^2 \)**:
\[ ( \sin x + \csc x )^2 = \sin^2 x + \csc^2 x + 2 \sin x \csc x \]
Since \( \csc x = \frac{1}{\sin x} \), we have:
\[ 2 \sin x \csc x = 2 \]
Thus,
\[ ( \sin x + \csc x )^2 = \sin^2 x + \csc^2 x + 2 \]

2. **Expand \( ( \cos x - \sec x )^2 \)**:
\[ ( \cos x - \sec x )^2 = \cos^2 x + \sec^2 x - 2 \cos x \sec x \]
Since \( \sec x = \frac{1}{\cos x} \), we have:
\[ -2 \cos x \sec x = -2 \]
Thus,
\[ ( \cos x - \sec x )^2 = \cos^2 x + \sec^2 x - 2 \]

3. **Expand \( ( \tan x + \cot x )^2 \)**:
\[ ( \tan x + \cot x )^2 = \tan^2 x + \cot^2 x + 2 \tan x \cot x \]
Since \( \tan x \cot x = 1 \), we have:
\[ 2 \tan x \cot x = 2 \]
Thus,
\[ ( \tan x + \cot x )^2 = \tan^2 x + \cot^2 x + 2 \]

### Step 2: Substitute back into the original expression
Now substituting back into the original expression:
\[ ( \sin^2 x + \csc^2 x + 2 ) + ( \cos^2 x + \sec^2 x - 2 ) - ( \tan^2 x + \cot^2 x + 2 ) \]

### Step 3: Combine the terms
Combining all the terms:
\[ \sin^2 x + \csc^2 x + 2 + \cos^2 x + \sec^2 x - 2 - \tan^2 x - \cot^2 x - 2 \]
This simplifies to:
\[ \sin^2 x + \csc^2 x + \cos^2 x + \sec^2 x - \tan^2 x - \cot^2 x \]

### Step 4: 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 rewrite:
\[ \csc^2 x + \sec^2 x = (1 + \cot^2 x) + (1 + \tan^2 x) = 2 + \tan^2 x + \cot^2 x \]

### Step 5: Substitute back
Now substituting back:
\[ 1 + (2 + \tan^2 x + \cot^2 x) - \tan^2 x - \cot^2 x \]
This simplifies to:
\[ 1 + 2 = 3 \]

### Final Result
Thus, the value of the expression is:
\[ \boxed{3} \]