Math Is Fun Forum

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?

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 52,337

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.

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 52,337

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.

Neela

Guest

Re: proving the identity

Thank you so much

Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 52,337

Re: proving the identity

Neela, welcome to the forum.

---

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.

Neela

Guest

Re: proving the identity

Thank you, Ganesh

See you around