Answer:
The given equation is:
Csc² x (1 + sin²x) = cot²x
To prove this equation, we will start with the left-hand side of the equation and simplify it using trigonometric identities.
LHS: Csc² x (1 + sin²x)
= 1/sin²x * (1 + sin²x) [Using the identity csc²x = 1/sin²x]
= 1 + sin²x/sin²x
= 1 + 1/cos²x [Using the identity sin²x + cos²x = 1]
= (cos²x + 1)/cos²x
= cot²x [Using the identity cot²x = cos²x/sin²x]
Therefore, LHS = RHS, which proves the given equation.