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Simplify cos expression
- Thread starter Chrizzle
- Start date
D
Deleted member 4993
Guest
Hey guys,
Having a little trouble simplifying this cos expression
cos4Θ - 6cos2Θ(1-cos2Θ)+(1-cos2Θ)2
Any help would be appreciated.
Cheers.
Hint:
1-cos2Θ = sin2Θ
D
Deleted member 4993
Guest
cos4Θ - 6cos2Θ(1-cos2Θ)+(1-cos2Θ)2
= cos4Θ - 6cos2Θ*sin2Θ + (sin2Θ)2
= cos4Θ + sin4Θ + 2cos2Θ*sin2Θ - 8cos2Θ*sin2Θ
continue.....
= cos4Θ - 6cos2Θ*sin2Θ + (sin2Θ)2
= cos4Θ + sin4Θ + 2cos2Θ*sin2Θ - 8cos2Θ*sin2Θ
continue.....
Hello, Chrizzle!
It does not simplify drastically, but I'll do my best.
Substituting \(\displaystyle 1-\cos^2\theta\:=\:\sin^2\theta\),
. . we have: .\(\displaystyle \cos^4\theta - 6\;\!\cos^2\theta\;\!\sin^2\theta + \sin^4\theta\)
. . . . . . . \(\displaystyle =\;\cos^4\theta + 2\;\!\cos^2\theta\;\!\sin^2\theta + \sin^4\theta - 8\;\!\cos^2\theta\;\!\sin^2\theta\)
. . . . . . . \(\displaystyle =\;\underbrace{(\cos^2\theta + \sin^2\theta)^2}_{\text{this is }1^2} - 8\;\!\cos^2\theta\;\!\sin^2\theta\)
. . . . . . . \(\displaystyle =\;1 - 8\;\!\cos^2\theta\;\!\sin^2\theta\)
. . . . . . . \(\displaystyle =\;1 - 2(4\;\!\sin^2\theta\;\!\cos^2\theta)\)
. . . . . . . \(\displaystyle =\;1 - 2(2\sin\theta\cos\theta)^2\)
. . . . . . . \(\displaystyle =\;1 - 2\;\!\sin^22\theta\)
It does not simplify drastically, but I'll do my best.
Chrizzle said:\(\displaystyle \text{Simplify: }\:\cos^4\theta - 6\;\!\cos^2\theta(1-\cos^2\theta) + (1-\cos^2\theta)^2\)
Substituting \(\displaystyle 1-\cos^2\theta\:=\:\sin^2\theta\),
. . we have: .\(\displaystyle \cos^4\theta - 6\;\!\cos^2\theta\;\!\sin^2\theta + \sin^4\theta\)
. . . . . . . \(\displaystyle =\;\cos^4\theta + 2\;\!\cos^2\theta\;\!\sin^2\theta + \sin^4\theta - 8\;\!\cos^2\theta\;\!\sin^2\theta\)
. . . . . . . \(\displaystyle =\;\underbrace{(\cos^2\theta + \sin^2\theta)^2}_{\text{this is }1^2} - 8\;\!\cos^2\theta\;\!\sin^2\theta\)
. . . . . . . \(\displaystyle =\;1 - 8\;\!\cos^2\theta\;\!\sin^2\theta\)
. . . . . . . \(\displaystyle =\;1 - 2(4\;\!\sin^2\theta\;\!\cos^2\theta)\)
. . . . . . . \(\displaystyle =\;1 - 2(2\sin\theta\cos\theta)^2\)
. . . . . . . \(\displaystyle =\;1 - 2\;\!\sin^22\theta\)