Given: tan(x) - cot(x) = 1, find the value of: tan(x)^6 + cot(x)^6
tan(x)^6 + cot(x)^6 = (tan(x) - cot(x))^6 + 6(tan(x) - cot(x))^5(tan(x) + cot(x))
= 1^6 + 6(1)^5(tan(x) + cot(x))
= 6(tan(x) + cot(x))
(tanx - cotx)^2 = tan^2x - 2tanx cotx + cot^2x = 1
tan^2x + cot^2x = 2
(tanx + cotx)^2 = tan^2x + 2tanx cotx + cot^2x = tan^2x + cot^2x + 1= 3
tanx + cotx = √2
Now,
tan^6x + cot^6x = (tan^3x + cot^3x)( tan^3x - cot^3x)
= (tanx+cotx)(tan^2x-tanx cotx+cot^2x)(tanx-cotx)(tan^2x+tanx cotx+cot^2x)
= (tanx+cotx)((tan^2x+cot^2x)^2-1)((tan^2x+cot^2x)^2+1)
= √2 (4-1)(4+1)
= 15√2
Hmmm. wolframalpha . com gets 18
Maybe you can catch my error