Prove that `cot^(4)theta+cot^(2)theta="cosec"^(4)theta-"cosec"^(2)theta`

Answer

To prove that \( \cot^4 \theta + \cot^2 \theta = \csc^4 \theta - \csc^2 \theta \), we will start with the right-hand side and manipulate it to show that it equals the left-hand side.

### Step 1: Start with the Right-Hand Side
We begin with the expression on the right-hand side:
\[ \csc^4 \theta - \csc^2 \theta \]

### Step 2: Factor Out \(\csc^2 \theta\)
We can factor out \(\csc^2 \theta\) from the right-hand side:
\[ \csc^4 \theta - \csc^2 \theta = \csc^2 \theta (\csc^2 \theta - 1) \]

### Step 3: Use the Pythagorean Identity
Recall the Pythagorean identity:
\[ \csc^2 \theta - 1 = \cot^2 \theta \]
Substituting this identity into our expression gives:
\[ \csc^2 \theta (\csc^2 \theta - 1) = \csc^2 \theta \cdot \cot^2 \theta \]

### Step 4: Substitute for \(\csc^2 \theta\)
We know that:
\[ \csc^2 \theta = 1 + \cot^2 \theta \]
Substituting this into our expression:
\[ \csc^2 \theta \cdot \cot^2 \theta = (1 + \cot^2 \theta) \cdot \cot^2 \theta \]

### Step 5: Expand the Expression
Now, we expand the expression:
\[ (1 + \cot^2 \theta) \cdot \cot^2 \theta = \cot^2 \theta + \cot^4 \theta \]

### Step 6: Rearrange the Terms
Rearranging gives us:
\[ \cot^4 \theta + \cot^2 \theta \]

### Conclusion
Thus, we have shown that:
\[ \csc^4 \theta - \csc^2 \theta = \cot^4 \theta + \cot^2 \theta \]
This proves that:
\[ \cot^4 \theta + \cot^2 \theta = \csc^4 \theta - \csc^2 \theta \]