Expert: Sherman D. - 12/3/2007

Question
can your show me the proof for

4((sinx)^6+(cosx)^6)= 4-3(sin2x)^2

Answer
4(sin(x)^6 + cos(x)^6)
4(sin(x)^6 + (cos(x)^2)^3)
4(sin(x)^6 + (1 - sin(x)^2)^3)

(1 - sin(x)^2)^3
(1 - sin(x)^2)(1 - sin(x)^2)(1 - sin(x)^2)
(1 - 2sin(x)^2 + sin(x)^4)(1 - sin(x)^2)
1 - sin(x)^2 - 2sin(x)^2 + 2sin(x)^4 + sin(x)^4 - sin(x)^6
1 - 3sin(x)^2 + 3sin(x)^4 - sin(x)^6

4(sin(x)^6 + (1 - sin(x)^2)^3)
4(sin(x)^6 + 1 - 3sin(x)^2 + 3sin(x)^4 - sin(x)^6)
4(1 - 3sin(x)^2 + 3sin(x)^4)
4(1 - 3sin(x)^2(1 - sin(x)^2))
4(1 - 3sin(x)^2cos(x)^2
4 - 12sin(x)^2cos(x)^2

this is the same as saying

4 - 3(2sin(x)cos(x) * 2sin(x)cos(x))

which would make this

4 - 3sin(2x)^2

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the way you had your problem would had been better written as

4(sin(x)^6 + cos(x)^6) = 4 - 3sin(2x)^2

if you have anymore problems like this, just let me know, and try to write them like the way i showed you.

for more info, go to www.math.com/tables/trig/identities.htm

sometimes it helps to rearrange the problem so that they work out the way you need them to work out.