Question

Solve cos^10 x-sin^10 x=1, 0≤x≤2π.

Answer 1

\[\large \cos ^{10}x-\sin ^{10}x= 1\] @phi

Answer 2

0 is a solution... cos0 =1, sin0 = 0

Answer 3

2pi is also a solution, since cos2pi = 1, sin2pi is zero.

There might be a not-too-difficult way to solve it algebraically, i thought about using a difference of two squares but idk if that helps. You can also use trig identities, but that also didn't seem helpful.

Answer 4

The rest are all complex solutions: http://www.wolframalpha.com/input/?i=+cos%5E10+x-sin%5E10+x%3D1

Answer 5

that helps a lot, thank-you so much. ♥ I will try a few more ways to solve it but I understand it better now.

Answer 6

There doesn't appear to be a simple algebraic way to solve it... not with the tenth powers.

Answer 7

Yeah.. that's true, looking at wolfram. I think 0 and 2π are the answers my teacher is looking for. thanks!

Answer 8

Wow agent, good job!!!! I expanded this equation out SOOOO LONG it's ridiculous. But I had used the identity \[\cos(2x)=1-2\sin ^{2}x\]
Good going @agent0smith :)

Answer 9

hehe, yeah with the tenth powers, i figured expanding was just going to leave you with a gigantic equation (which still wouldn't be easy to solve).

Answer 10

And you were absolutely correct. :|

Answer 11

Thanks for posting the problem @hymnets That was a bit of brain fun and trig review :)

Answer 12

thanks for helping out and spending time on it, I really appreciate it. : ) I have this math summative which consists of a sheet of 13 very difficult questions that I have to memorize so there's a lot where this came from, haha... (wincing)

Answer 13

Off hand, I don't know how to solve such an ugly equation. However, I would plot cos
then "modify it" to show cos^10 (it will always be positive and have sharper peaks)
do the same for sin^10
sketch the curve (roughly) and see that there are only a few points where this curve is 1
at 0, pi, and 2pi