)Miike012 said:
The given equation to prove is Tan^2 - sin^2 = sin^4*Sec^4.
The identities used to prove this equation are the Pythagorean identity for tangent, Tan^2 + 1 = Sec^2, and the double angle identity for sine, sin(2x) = 2sin(x)cos(x).
To start the proof, we can rewrite the left side of the equation using the Pythagorean identity for tangent. This gives us Sec^2 - sin^2 = sin^4*Sec^4.
The next step is to use the double angle identity for sine to expand the right side of the equation. This gives us Sec^2 - sin^2 = (sin^2)^2*(Sec^2)^2.
Finally, we can use the Pythagorean identity for sine, sin^2 + cos^2 = 1, to simplify the right side of the equation. This gives us Sec^2 - sin^2 = (1 - cos^2)^2*(Sec^2)^2. Then, by distributing and simplifying, we can show that the left and right sides are equal, thus proving the given equation.