Question

prove:cos^3 x-sin^3 x = (cosx -sinx)(1+1/2 sin 2x)

Answer 1

\(cos^3 x-sin^3 x = (cosx -sinx)(1+1/2 sin 2x)\).. do RHS first.

Answer 2

okay

Answer 3

then what

Answer 4

sorry for delay, i fell into the den of many Q-cats... lol

Answer 5

its okay @koikkara

Answer 6

So Koi was suggesting that we apply our `Difference of Cubes` Formula :)\[\large\rm a^3-b^3=(a-b)(a^2+ab+b^2)\]

Answer 7

Do you see how this matches our formula?\[\large\rm \cos^3 x-\sin^3 x = (\cos x-\sin x)(\cos^2x+\cos x~\sin x+\sin^2x)\]

Answer 8

Oh he said do right hand side first, I guess he was suggesting something else :))
hehe

Answer 9

I'm showing the path if we start on left side.

Answer 10

well i started with the right side because it is more complex

Answer 11

Hmm I think right hand side is going to be a lot more difficult :d
sec thinking.

Answer 12

okay so what do i do after applying the difference of cubes formula

Answer 13

Do you notice anything going on right here?\[\large\rm \cos^3 x-\sin^3 x = (\cos x-\sin x)(\color{orangered}{\cos^2x}+\cos x~\sin x\color{orangered}{+\sin^2x})\]It's one of your Pythagorean Identities.

Answer 14

yes \[\sin^2x \theta+\cos^2 \theta =1\]

Answer 15

Ok that brings us here,\[\large\rm \cos^3 x-\sin^3 x = (\cos x-\sin x)(\color{orangered}{1}+\cos x~\sin x)\]

Answer 16

Recall your Double-Angle Formula for Sine? :)

Answer 17

We'll have to be a little sneaky to make it work for us.

Answer 18

yes i do

Answer 19

\[\sin(2x)_2sinxcosx\]

Answer 20

Hehe something got a little mixed up there :)
I think you meant to write this:\[\large\rm \sin(2x)=2\sin x \cos x\]Let's fiddle around with this identity just a tad bit.
Let's divide both sides by 2,\[\large\rm \frac{1}{2}\sin(2x)=\sin x \cos x\]Do you see how that will help us?

Answer 21

yess

Answer 22

\[\large\rm \color{orangered}{\frac{1}{2}\sin(2x)=\sin x \cos x}\]And we have this orange thing in our expansion,\[\large\rm \cos^3 x-\sin^3 x = (\cos x-\sin x)(1+\color{orangered}{\cos x\sin x})\]\[\large\rm \cos^3 x-\sin^3 x = (\cos x-\sin x)\left(1+\color{orangered}{\frac{1}{2}\sin(2x)}\right)\]So I guess that's our final step, ya?
Yay we did it \c:/

Answer 23

yay its proved