Question

(cotx−1)^2



cot^2x+2cscx

csc^2x−2cotx

cot^2x−2cscx
csc^2x+2cotx

Answer 1

To simplify the expression \((\cot x - 1)^2\) and relate it to the given options, let's first expand \((\cot x - 1)^2\):

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

Next, we can express \(\csc^2 x\) in terms of \(\cot^2 x\) using the identity \(\csc^2 x = 1 + \cot^2 x\):

1. **Substituting for \(\csc^2 x\)**:

\[
\csc^2 x = 1 + \cot^2 x
\]

2. **Rearranging the expanded form**:

From \((\cot x - 1)^2 = \cot^2 x - 2\cot x + 1\), we replace 1 with \(\csc^2 x - \cot^2 x\):

\[
\cot^2 x - 2\cot x + (\csc^2 x - \cot^2 x) = -2\cot x + \csc^2 x
\]

This shows:

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

Thus, we can see that:

\[
(\cot x - 1)^2 = \csc^2 x - 2 \cot x
\]

Finally, among the provided options, the suitable answer is:

\[
\csc^2 x - 2 \cot x
\]

So the correct matching expression is:

\[
\csc^2 x - 2 \cot x
\]