SOLUTION: Question 624957: Verify the trig identity: sin^4(x)+cos^4 (x)=3/4 + 1/4 cos(4x). How about this solution (different and faster than from jsmallt9): sin^4(x)+cos^4(x) = ((sin^2

Algebra ->  Trigonometry-basics -> SOLUTION: Question 624957: Verify the trig identity: sin^4(x)+cos^4 (x)=3/4 + 1/4 cos(4x). How about this solution (different and faster than from jsmallt9): sin^4(x)+cos^4(x) = ((sin^2      Log On


   

Question 624980: Question 624957: Verify the trig identity: sin^4(x)+cos^4 (x)=3/4 + 1/4 cos(4x).
How about this solution (different and faster than from jsmallt9):
sin^4(x)+cos^4(x) = ((sin^2(x) + cos^2(x)))^2 - 2sin^2(x)cos^2(x) =
1 - 2sin^2(x)cos^2(x) = 1 - (1/2)sin^2(2x) = 1 - (1 - cos(4x))/4 =
1 - 1/4 + cos(4x)/4 = 3/4 + cos(4x)/4 Done
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
This solution works, too.

The main reason it appears faster than my solutions is that many intermediate steps are being omitted. If I had done this my first solution would look like:

2%2F4%2B%282cos%5E2%282x%29%29%2F4+=+3%2F4+%2B+%281%2F4%29cos%284x%29
3%2F4%2B%281%2F4%29cos%284x%29+=+3%2F4+%2B+%281%2F4%29cos%284x%29
And we're finished!

And the second solution would be:
%28sin%5E2%28x%29%29%5E2%2B%28cos%5E2%28x%29%29%5E2=3%2F4+%2B+%281%2F4%29cos%284x%29

%281%2F2%29cos%5E2%282x%29+%2B+1%2F2+=+3%2F4+%2B+%281%2F4%29cos%284x%29


%281%2F4%29cos%284x%29+%2B+3%2F4+=+3%2F4+%2B+%281%2F4%29cos%284x%29