Proving 1 CosecA/SecA SecA/1 CosecA = 2Cosec^3A(SecA-1)

To prove the given equation, we will simplify both sides of the equation independently and then equate them to check if they are equal.

Step 1: Simplifying the left-hand side (LHS) of the equation

We are given the equation: 1 CosecA/SecA SecA/1 CosecA = 2Cosec^3A(SecA-1)

Let's start by simplifying the left-hand side (LHS) of the equation:

1 CosecA/SecA can be written as (1/1) * (CosecA/SecA) = CosecA/SecA

Similarly, SecA/1 CosecA can be written as (SecA/1) * (1/CosecA) = SecA/CosecA

Therefore, the left-hand side (LHS) of the equation becomes: CosecA/SecA SecA/CosecA

Step 2: Simplifying the right-hand side (RHS) of the equation

Now let's simplify the right-hand side (RHS) of the equation:

2Cosec^3A(SecA-1) can be expanded as 2 * (CosecA * CosecA * CosecA) * (SecA - 1)

Using the identity Cosec^2A = 1 + Cot^2A, we can rewrite CosecA as 1/CosecA = 1/(1 + Cot^2A)

Substituting the value of CosecA in the equation, we get:

2 * (1/(1 + Cot^2A)) * (1/(1 + Cot^2A)) * (1/(1 + Cot^2A)) * (SecA - 1)

Simplifying further, we get:

2/(1 + Cot^2A)^3 * (SecA - 1)

Step 3: Equating LHS and RHS

Now that we have simplified both sides of the equation, let's equate them and check if they are equal:

CosecA/SecA SecA/CosecA = 2/(1 + Cot^2A)^3 * (SecA - 1)

To simplify further, we can cross multiply:

(CosecA * CosecA * SecA * SecA) = 2 * (SecA - 1) * (1 + Cot^2A)^3

Using the identity Cosec^2A = 1 + Cot^2A, we can rewrite the equation as:

(SecA * SecA) = 2 * (SecA - 1) * (1 + Cot^2A)^3

Expanding (1 + Cot^2A)^3, we get:

(SecA * SecA) = 2 * (SecA - 1) * (1 + Cot^2A) * (1 + Cot^2A) * (1 + Cot^2A)

Simplifying further, we have:

(Sec