prove the identity

Question

prove sin^4(y)-cos^4(y)= 2sin^2(x) -1

i think you would start on the left hand side but im not sure how to start or why the varibles change. thanks in advance for the help!

Answer 1

This is written in two different variables (left side is "y" and right side is "x"). If you are trying to prove an identity, they would all have the same variable. I'm going to assume this is a typo and I'll use "x" on both sides:

sin⁴(x) - cos⁴(x) = 2sin²(x) - 1 [factor the left side as a difference of 2 squares to get:

(sin²(x) + cos²(x))(sin²(x) - cos²(x)) = 2sin²(x) - 1 [using the Pythagorean ID, sin² + cos² = 1 we get:

(sin²(x) - cos²(x)) = 2sin²(x) - 1 [solving the ID sin² + cos² = 1 for cos²: cos² = 1 - sin² and subbing:

(sin²(x) - [1 - sin²(x)] = 2sin²(x) - 1 [distributing and simplifying we get:

2sin²(x) - 1 = 2sin²(x) - 1

Answer 2

I'm using 30°.

(sin30)^4-(cos30)^4=2(sinx)^2-1

.5^4 - .866^4 = " " "

.0625 - .562 = " " "

-.5 = The best I can do is x=360=-1.