To prove the given equation cosecA - 1/cosecA = cotA - cosA/cotA * cosA, we need to simplify both sides of the equation and show that they are equal.

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

cosecA - 1/cosecA

To simplify this expression, we need to find a common denominator for the two terms. The common denominator is cosecA, so we can rewrite the expression as:

(cosecA * cosecA - 1) / cosecA

Now, we can simplify the numerator:

cosecA * cosecA - 1

Using the identity cosecA = 1/sinA, we can rewrite the numerator as:

(1/sinA) * (1/sinA) - 1

Simplifying further, we get:

1/sin^2A - 1

Now, let's simplify the right-hand side of the equation:

cotA - cosA/cotA * cosA

First, let's simplify the numerator:

cotA - cosA

Using the identity cotA = cosA/sinA, we can rewrite the numerator as:

cosA/sinA - cosA

Now, we need to find a common denominator for the two terms. The common denominator is sinA, so we can rewrite the expression as:

(cosA - cosA * sinA) / sinA

Simplifying further, we get:

cosA(1 - sinA) / sinA

Now, let's simplify the denominator:

cotA * cosA

Using the identity cotA = cosA/sinA, we can rewrite the denominator as:

cosA * cosA/sinA

Simplifying further, we get:

cos^2A / sinA

Now, let's rewrite the right-hand side of the equation with the simplified numerator and denominator:

cosA(1 - sinA) / sinA * cos^2A / sinA

Multiplying these fractions together, we get:

cosA(1 - sinA) * cos^2A / sinA^2

Expanding the numerator, we get:

cos^3A - cos^2A * sinA

Now, let's simplify both sides of the equation and show that they are equal:

Left-hand side: 1/sin^2A - 1

Right-hand side: cos^3A - cos^2A * sinA

To simplify further, we can use the identity sin^2A = 1 - cos^2A:

Left-hand side: 1/(1 - cos^2A) - 1

Right-hand side: cos^3A - cos^2A * sinA

Now, let's find a common denominator for the left-hand side:

(1 - cos^2A)/(1 - cos^2A) - 1

Simplifying further, we get:

(1 - cos^2A - (1 - cos^2A))/(1 - cos^2A)

This simplifies to:

0/(1 - cos^2A)

Since the numerator is 0, the left-hand side of the equation is equal to 0.

Now, let's simplify the right-hand side:

cos^3A - cos^2A * sinA

Using the identity

SIMPLIFY RHS into cos and sin then divide the num• and denom•by cosA