Hannah_Ahn
prove the identity
csc theta over tan theta + cot theta = cos theta
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Zed
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\[\frac{ \frac{1}{sin {\theta}}}{tan{\theta}}+\frac{1}{tan{\theta}}=cos{\theta}\]
Hannah_Ahn
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I am sorry, how did that happen?
Zed
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Sorry I will write it up first and post it with steps. Just a moment
Hannah_Ahn
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Sweet, thanks!
Yifan12879
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does it mean that (1/sin x)^2+(1/sin x)=1?
Yifan12879
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if you divide the cos theta from both sides
Zed
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Yes correct
Hannah_Ahn
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I still would appreciate to see the steps plz :s
Yifan12879
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but why? let x=pi/6 then 4+2=1??
Zed
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yes I'm having that issue too. Are you sure it's written down right?
Hannah_Ahn
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\[\csc \theta \over \tan \theta +\cot \theta\]
=
[\cos \theta\]
Zed
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I see now, next time put the tan theta + cot theta in brackets to make it clearer.
Yifan12879
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multiply the LHS by sinx cosx
Hannah_Ahn
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gotcha. sorry about that.
Yifan12879
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i mean csc x /(tan x + cot x)= cscx sinx cosx / (tanx sinx cosx+cotx sinx cosx)=cosx/(sin^2 x+cos^2 x)=cosx
Yifan12879
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im sorry it should be theta...
Hannah_Ahn
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why did you multiply each sides by sinx cosx? how could u tell?
Zed
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\[\frac{1/sin{\theta}}{tan{\theta}+1/tan{\theta}}=cos{\theta}\]\[1/sin{\theta}=tan{\theta}cos{\theta}+cos{\theta}/tan{\theta}\]\[1/sin{\theta}=sin{\theta}+cos^2{\theta}/sin{\theta}\]\[1/sin{\theta}=sin^2{\theta}/sin{\theta}+cos^2{\theta}/sin{\theta}\]\[1/sin{\theta}=1/sin{\theta}\]\[1=1\]
Zed
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Here's a different way to go about it :)
Yifan12879
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because there are sinx cosx in the denominator
Hannah_Ahn
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my teacher doesn't let me combite the left side and the right side.. in this case i would need to make the left side cos theta. :S
Hannah_Ahn
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zed; yur way is something to think about, and it's cool.
yet i won't b able to use your method
Yifan12879
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@Zed you shouldnt give out a prove like this, you should write LHS=...=....=...=RHS
Hannah_Ahn
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yifian; i don't really get how you got there.. :(
Hannah_Ahn
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i can't figure it out
Yifan12879
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which step?
Hannah_Ahn
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is there other way to do it?
Zed
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Thanks Yifan, I haven't done proof in a long time.
\[LHS = \frac{1/sinx}{sinx/cosx+cosx/sinx}\]
\[=\frac{1/sinx}{(sin^2x+cos^2x)/sinxcosx}\]\[=\frac{1/sinx}{1/sinxcosx}\]\[=\frac{1}{cosx}\]\[=RHS\]
Zed
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sorry that second last line was supposed to be cosx
Yifan12879
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yeah that is a good prove except that you substitute theta for x..
Zed
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yeah i got tired of writing theta out
Zed
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thanks Yifan
Hannah_Ahn
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1 over cos x isn't same as cos x
Zed
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It was a typo
Hannah_Ahn
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i got that too.
Hannah_Ahn
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well thanks for your time :)
Hannah_Ahn
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i still don't get it . for some reason, im very confused. i will ask my teacher and I will let you know if there is a simpler way to do it if you are interested :P g'nite.