(1-cos²θ)(1+cos²θ)=2sin²θ-sin⁴θ. Need to show this is an identity.

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(1-cos²θ)(1+cos²θ)=2sin²θ-sin⁴θ. Need to show this is an identity.

Mathematics
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\[\sin^2\theta+\cos^2\theta=1\]
use this wherever you see 1 then simplify it till you get your right hand side equation
best bet may be to write first as \[1-cos^4(\theta)\] then replace \[cos^4(\theta)\] by \[(1-sin^2(\theta))^2\] then multiply out and i believe this gives it to you.

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will work it out if you like. it is algebra from there on in.
oh my, could you work it out?
sure i will multiply out \[(1-sin^2(\theta))^2\] but it is just the same as \[(1-x^2)^2\] with x replaced by sine. \[(1-x^2)^2=1-2x^2+x^4\] so \[(1-sin^2(\theta))^2=1-2sin^2(\theta)+sin^4(\theta)\]\]
don't forget that originally you have \[1-cos^4(\theta)\] so now you have \[1-(1-2sin^2(\theta)+sin^4(\theta))=2sin^2(\theta)-sin^4(\theta)\]
is that enough detail? if not i let me know.
i don't know why, but i just can't see it
\[(\sin^2\theta+\cos^2\theta-\cos^2\theta)(\sin^2\theta+\cos^2\theta+\cos^2\theta)=(\sin^2\theta)(sina^2\theta+2\cos^2\theta)=\sin^4\theta+2\sin^2\theta(\cos^2\theta)=\sin^4\theta+2(\sin^2\theta)(1-\sin^2\theta)=sina^4\theta+2\sin^\theta-2\sin^4\theta=2\sin^2\theta-\sin^4\theta\]
ok lets to slowly. first of all, the left hand side of the equation is \[(1-cos^2(\theta))(1-cos^2(\theta))\]
so our first job is to multiply this out.
is it clear that this is the same as multiplying out \[(1-x)(1+x)\]?
yes, when you write it like that
\[or rather (x-x^2)(1+x^2)\]
typo sorry
ok so lets multiply out \[(1-cos^2(\theta))(1+cos^2(\theta))\]
\[(1-x^2)(1+x^2)=1-x^4\] so\[((1-cos^2(\theta))(1+cos^2(\theta))=1-cos^4(\theta)\] so far so good?
yes
ok now you recall that \[sin^2(\theta)+cos^2(\theta)=1\] so \[cos^2(\theta)=1-sin^2(\theta)\]
and of course \[cos^4(\theta)=(cos^2(\theta))^2\] so replace \[cos^2(\theta)\]by \[(1-sin^2(\theta))\] in this expression to get \[1-(1-sin^2(\theta))^2\]
ok?
oh, ok
now multiply out and you get your answer exactly. this is like \[1-(1-x^2)^2=1-(1-2x^2+x^4)=2x^2-x^4\] with x replaced by sine.
ooohhh
clear or no?
it is when you replace the sin/cos with the 'x'. easier to see
yes of course. the only little bit of trig here was replacing \[cos^2(\theta)\]by \[(1-sin^2(\theta)\] every other step was algebra. multiply, collect like terms etc.
thanks....
wanna do the next one?
i'm not sure where to start

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