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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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 45,966

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


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 45,966

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


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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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

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 45,966

Re: proving the identity

Neela, welcome to the forum.

welcome1.gif


It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

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

Neela
Guest

Re: proving the identity

Thank you, Ganesh

See you around smile

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