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#1 2006-03-17 02:12:52

Neela
Guest

proving the identity

prove the identity:

(sin x + cos x) (1 - sin x cos x) = sin^3 x + cos^3 x

and

(sin 2x) /(1 + cos 2x) = tan x

anyone?

#2 2006-03-17 02:53:52

ganesh
Moderator

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Re: proving the identity

The second one seems simpler, I shall start with that.
Sin2x/(1+Cos2x)=tanx
Sin2x=SinxCosx+CosxSinx=2SinxCosx
Cos2x = Cos(x+x)= Cosx - Sinx
Therefore,
LHS=2SinxCosx/(1+Cosx - Sinx)
Since 1- Sinx=Cosx,
LHS=2SinxCosx/2Cosx = Sinx/Cosx=tanx=RHS
q.e.d


Character is who you are when no one is looking.

#3 2006-03-17 03:09:14

ganesh
Moderator

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Re: proving the identity

The first problem given is
(sin x + cos x) (1 - sin x cos x) = sin^3 x + cos^3 x.
Lets simplify the LHS...
(Sinx+Cosx)(1-SinxCosx)
=Sinx-SinxCosx+Cosx-SinxCosx
=Sinx-SinxCosx+Cosx-SinxCosx (Rearranging the terms).
=Sinx(1-Cosx)+Cosx(1-Sinx)
Since 1-Cosx=Sinx  and 1-Sinx=Cosx,
LHS=Sinx(Sinx)+Cosx(Cosx)= Sinx+Cosx=RHS
q.e.d


Character is who you are when no one is looking.

#4 2006-03-17 03:14:23

Neela
Guest

Re: proving the identity

Thank you so much smile

#5 2006-03-17 03:16:36

ganesh
Moderator

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Re: proving the identity

Neela, welcome to the forum.

http://www.geocities.com/aishwarya_82/images/welcome1.gif


Character is who you are when no one is looking.

#6 2006-03-17 03:25:07

Neela
Guest

Re: proving the identity

Thank you, Ganesh

See you around smile

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