Prove the identity.

(sin^2x) / (1-cosx) ^2 = (1+cosx) / (1-cosx)

Step-by-step explanation:

We consider the (sin²x) / (1-cos x) ² as left hand side (LHS)

While (1+cosx) / (1-cosx) is RHS (Right hand side)

Before proceeding with the equation, we must know some trigonometric and algebraic corollary such as-

Sin²x = 1-cos²x

a²-b² = (a + b) * (a - b)

(a-b) ² = (a - b) * (a-b)

Using these corollaries to our advantage in solving our problem-

From 1 sin²x can be written as 1-cos²x

Similarly, 1-cos²x can be written as 1²-cos²x (in the form of a²-b² and then using the formula 2)

Thus, sin²x can be finally written as (1+cos x) * (1-cos x)

Similarly (1-cos x) ² can be written as (1-cos x) * (1-cos x)

Putting the above two steps in the LHS equation-

(1+cos x) * (1-cos x) / (1-cos x) * (1-cos x) (common term (1-cos x) gets cancelled out)

LHS = (1+cos x) / (1-cos x)

Hence LHS=RHS