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#1 2006-03-16 03:12:52

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 2006-03-16 03:53:52

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,206

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)= Cos²x - Sin²x
Therefore,
LHS=2SinxCosx/(1+Cos²x - Sin²x)
Since 1- Sin²x=Cos²x,
LHS=2SinxCosx/2Cos²x = Sinx/Cosx=tanx=RHS
q.e.d


Character is who you are when no one is looking.

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#3 2006-03-16 04:09:14

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,206

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-Sin²xCosx+Cosx-SinxCos²x
=Sinx-SinxCos²x+Cosx-Sin²xCosx (Rearranging the terms).
=Sinx(1-Cos²x)+Cosx(1-Sin²x)
Since 1-Cos²x=Sin²x  and 1-Sin²x=Cos²x,
LHS=Sinx(Sin²x)+Cosx(Cos²x)= Sin³x+Cos³x=RHS
q.e.d


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#4 2006-03-16 04:14:23

Neela
Guest

Re: proving the identity

Thank you so much smile

#5 2006-03-16 04:16:36

ganesh
Moderator
Registered: 2005-06-28
Posts: 15,206

Re: proving the identity

Neela, welcome to the forum.

welcome1.gif


Character is who you are when no one is looking.

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#6 2006-03-16 04:25:07

Neela
Guest

Re: proving the identity

Thank you, Ganesh

See you around smile

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