What is the value of `(1+cotA)^(2)+(1-cotA)^(2)`?

Options

A. `2cosec^(2)A`
B. `2sec^(2)A`
C. `1-2cosec^(2)A`
D. `1-2sec^(2)A`

Answer

To solve the expression \( (1 + \cot A)^2 + (1 - \cot A)^2 \), we can follow these steps:

### Step 1: Expand the squares
Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \) and \( (a - b)^2 = a^2 - 2ab + b^2 \), we can expand both terms:

\[ (1 + \cot A)^2 = 1^2 + 2 \cdot 1 \cdot \cot A + (\cot A)^2 = 1 + 2 \cot A + \cot^2 A \]

\[ (1 - \cot A)^2 = 1^2 - 2 \cdot 1 \cdot \cot A + (\cot A)^2 = 1 - 2 \cot A + \cot^2 A \]

### Step 2: Combine the expanded expressions
Now, we add the two expanded expressions together:

\[ (1 + \cot A)^2 + (1 - \cot A)^2 = (1 + 2 \cot A + \cot^2 A) + (1 - 2 \cot A + \cot^2 A) \]

### Step 3: Simplify the expression
Combining like terms:

\[ = 1 + 2 \cot A + \cot^2 A + 1 - 2 \cot A + \cot^2 A \]

The \( +2 \cot A \) and \( -2 \cot A \) cancel each other out:

\[ = 1 + 1 + \cot^2 A + \cot^2 A = 2 + 2 \cot^2 A \]

### Step 4: Factor the expression
We can factor out the common term:

\[ = 2(1 + \cot^2 A) \]

### Conclusion
Thus, the value of \( (1 + \cot A)^2 + (1 - \cot A)^2 \) is:

\[ \boxed{2(1 + \cot^2 A)} \]

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