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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?
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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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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Thank you so much
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.
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Thank you, Ganesh
See you around
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