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**Neela****Guest**

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?

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 23,044

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 is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

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

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 23,044

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 is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

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

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**Neela****Guest**

Thank you so much

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 23,044

Neela, welcome to the forum.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

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

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**Neela****Guest**

Thank you, Ganesh

See you around

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