Problem:
Given that x = tan A sin A and y = tan A - sin A, prove that x^2 - y^2 = 4√xy.

Solution:

Step 1: Express x and y in terms of sin A and tan A.

Given:
x = tan A sin A
y = tan A - sin A

Step 2: Square both sides of the equations to eliminate the square root.

Squaring x:
x^2 = (tan A sin A)^2
x^2 = tan^2 A sin^2 A
x^2 = (sin^2 A / cos^2 A) sin^2 A
x^2 = sin^4 A / cos^2 A

Squaring y:
y^2 = (tan A - sin A)^2
y^2 = tan^2 A - 2tan A sin A + sin^2 A
y^2 = (sin^2 A / cos^2 A) - 2(sin A / cos A) sin A + sin^2 A
y^2 = sin^2 A - 2sin^2 A + cos^2 A
y^2 = -sin^2 A + cos^2 A
y^2 = cos^2 A - sin^2 A

Step 3: Rearrange the equations to prepare for substitution.

Rearranging x^2:
x^2 = sin^4 A / cos^2 A
x^2 = (sin^2 A)^2 / (cos^2 A)

Rearranging y^2:
y^2 = cos^2 A - sin^2 A
y^2 = (cos^2 A - sin^2 A)

Step 4: Substitute the values of x^2 and y^2 into the expression x^2 - y^2.

x^2 - y^2 = (sin^2 A)^2 / (cos^2 A) - (cos^2 A - sin^2 A)
x^2 - y^2 = sin^4 A / cos^2 A - cos^2 A + sin^2 A

Step 5: Simplify the expression.

Using the identity sin^2 A + cos^2 A = 1, we can rewrite sin^2 A as 1 - cos^2 A.

x^2 - y^2 = sin^4 A / cos^2 A - cos^2 A + sin^2 A
x^2 - y^2 = (1 - cos^2 A)^2 / cos^2 A - cos^2 A + sin^2 A
x^2 - y^2 = (1 - 2cos^2 A + cos^4 A) / cos^2 A - cos^2 A + sin^2 A
x^2 - y^2 = 1 - 2cos^2 A + cos^4 A - cos^4 A - cos^2 A + sin^2 A
x^2 - y^2 = 1 - 3cos^2 A - cos^2 A + sin^2 A
x^2 - y^2 = 1 - 4cos^

I think your q is not appropirate