Question

What is the sum of all values of x (in degrees) satisfying sin^6 x−cos^6 x=1, where 0≤x≤360?

Answer 1

\[\cos^{6} x = (1-\sin^{2}x)^{3} = 1-3\sin^{2} x +3\sin^{4} x -\sin^{6} x\]
\[\rightarrow 2\sin^{6} x -3\sin^{4} x +3\sin^{2}x -2 = 0\]
let u = sin(x)
\[2u^{6}-3u^{4}+3u^{2} -2 = 0\]
by graphing you can see roots of +-1
it factors:
\[(u^{2}-1)(2u^{4}-u^{2}+2) = 0\]
\[u^{2} = \sin^{2}x = 1\]
\[\sin x = \pm 1\]
\[x = 90, 270\]
sum is 360

Answer 2

what about Sin^12x-Cos^12x =1

Answer 3

its done the same way...expand cos^12 in terms of sin
then find roots of the 12th degree polynomial