SOLUTION: Prove or disprove: cos^6(x)=-2+3cos^2(x)+3sin^4(x)-sin^6(x) Can you please show me the steps for solving this problem.

Algebra ->  Trigonometry-basics -> SOLUTION: Prove or disprove: cos^6(x)=-2+3cos^2(x)+3sin^4(x)-sin^6(x) Can you please show me the steps for solving this problem.      Log On


   

Question 1151556: Prove or disprove: cos^6(x)=-2+3cos^2(x)+3sin^4(x)-sin^6(x)
Can you please show me the steps for solving this problem.
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
cos%5E6%28x%29=-2%2B3cos%5E2%28x%29%2B3sin%5E4%28x%29-sin%5E6%28x%29

%28cos%5E2%28x%29%5E%22%22%29%5E3=-2%2B3cos%5E2%28x%29%2B3sin%5E4%28x%29-sin%5E6%28x%29

Substitute 1-sin²(x) for cos²(x)





To uncomplicate matters, let y = sin²(x)

%281-y%5E%22%22%29%5E3=-2%2B3%281-y%5E%22%22%29%2B3%28y%5E%22%22%29%5E2-%28y%5E%22%22%29%5E3

Simplify

1-3y%2B3y%5E2%2By%5E3=-2%2B3-3y%2B3y%5E2-y%5E3

Simplify further

1-3y%2B3y%5E2-y%5E3=1-3y%2B3y%5E2-y%5E3

Both sides are identical, so the problem is proved.
It's OK to work with both sides of an identity as long
as you don't change the value of either side, by adding,
subtracting, multiplying, or dividing to, from, or by both
sides.

Edwin