Question

prove the identity

1/4(cos(3(theta)=cos^3(theta)-3/4(cos(theta)

Answer 1

Sorry, but could you use equation editor to make it easier to read?

Answer 2

yeh i will try sorry about it

Answer 3

so easyit is

Answer 4

just in one step

Answer 5

simplify the rhs by the identity

Answer 6

simplify cos 3 theta

Answer 7

\[\frac{ 1 }{ 4 }\cos 3\theta=\cos^3\theta-\frac{ 3 }{ 4 }\cos \theta\]

Answer 8

cos3theta=4 cos^3 theta-3cos theta

Answer 9

if u simplify u will directly get the answer

Answer 10

could cos(3theta)=cos(2theta)+cos(theta)?

Answer 11

\[\cos(3x)\ne \cos(2x) +\cos(x)\]
You see from what you're trying to prove is 4cos^3 theta - 3 cos(theta)...

Answer 12

yes

Answer 13

then what is gonna be cos(3theta)?

Answer 14

4 cos^3 theta- 3 cos theta

Answer 15

this is an identity

Answer 16

ah! i get it now.. but how d oi prove the identity..

Answer 17

The derivation of this identity involves Euler's formula and complex numbers...I can explain I guess...

Answer 18

ya

Answer 19

now all the proofs of the identities are interrelated now u'll b asking for all those a what

Answer 20

Okay:
\[\cos(3x)=\Re (e^{3ix})=\Re ((e^{ix})^3)=\Re ((\cos(x)+i \sin(x))^3)=\]
\[=\Re(\cos^3(x)+i (3 \sin(x) \cos^2(x)-\sin^3(x))-3 \sin^2(x) \cos(x)) \]
\[=\cos^3(x)-3 \sin^2(x)\cos(x)=\cos^3(x)-3(1-\cos^2(x))\cos(x)=4\cos^3(x)-3\cos(x)\]
\[=\cos^3(x)-3(1-\cos^2(x))\cos(x)=4\cos^3(x)-3\cos(x)\]

Answer 21

If you want the sin(3x) one then take the imaginary part:
\[\sin(3x)=3\sin(x)\cos^2(x)-\sin^3(x))=3\sin(x)(1-\sin^2(x))-\sin^3(x)\]
\[=3\sin(x)-4\sin^3(x)\]

Answer 22

\[LHS=\frac{\cos(x+2x)}{4}=\frac{cosxcos2x-sinxsin2x}{4} \]
\[=\frac{cosx(\cos^2x-\sin^2x)-sinx(2sinxcosx)}{4} \]
\[=\frac{\cos^3x-cosxsin^2x-2\sin^2xcosx}{4} \]
\[=\frac{\cos^3x-3\sin^2xcosx}{4} \]

\[=\frac{\cos^3-3(1-\cos^2x)cosx}{4} \]
\[=\frac{\cos^3x-3cosx+3\cos^3x}{4} \]
\[=\frac{4\cos^3x-3cosx}{4} = \cos^3x - \frac{3cosx}{4}\]
.: LH=RHS

Answer 23

i am closing it sam bye
i got the answer lol

Answer 24

wtf my answer good

Answer 25

i know i got the answer
lol did you work it out?

Answer 26

yeah look at mine

Answer 27

i did lol you are so pro man haha

Answer 28

did u get same as mine?