Prove $ \sin^4(x)$ identity using $z=cis(x)?$

Question

I cannot prove the $\sin^4(x)$ identity using $z= cis(x)$. I know that you have to use de Moivre's theorem and compare the real values of $z^4$ but I am stuck at this step: $$\sin^4(x)= -\cos^4(x) + 6\sin^2(x)\cos^2(x) + \cos(4x)$$

The identity is $\sin^4(x) = 1/8(\cos(4x) - 4\cos(2x) + 3)$

the sin^4(x) identity *Which* identity? Be sure to spell out what you mean to really ask. — dxiv, Mar 11 '18 at 07:00
Use Moivre's theorem on one side and the Binomial theorem on the other. $\cos(4x)+i\sin(4x)=(\cos(x)+i\sin(x))^4=\cos^4(x)+4\cos^3(x)i\sin(x)+6\cos^2(x)i^2\sin^2(x)+4\cos(x)i^3\sin^3(x)+\sin^4(x)$. — YAlexandrov, Mar 11 '18 at 07:02
@VarunJahagirdar Click edit under your question and add it there, not as a comment. — dxiv, Mar 11 '18 at 07:05
it is the same question but they have not answered the question using complex numbers — Varun Jahagirdar, Mar 11 '18 at 07:09
They have answered the question using complex numbers, you can just replace $e^{i\theta} = cis(\theta)$ and you'd have the same answer. So I'd say this is probably a duplicate. — user2662833, Mar 11 '18 at 07:13
@VarunJahagirdar MPW’s answer certainly uses complex numbers. — Macavity, Mar 11 '18 at 07:15
Yes that is true... but I haven't yet learnt the use of Euler's number sorry — Varun Jahagirdar, Mar 11 '18 at 07:26
@VarunJahagirdar Please remember that you can choose an aswer among the given if the OP is solved, more details here https://meta.stackexchange.com/questions/5234/how-does-accepting-an-answer-work — user, Mar 12 '18 at 23:56

user · Answer 1 · 2018-03-11T07:36:21.950

1

Note that

$$cos 4x+i\sin4x=(\cos x+i\sin x)^4$$

then use

$$(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$$

which leads to

$\sin {4x}= 4\cos^3 x\sin x-4\cos x\sin^3x$
$\cos 4x= \cos^4 x-6\cos^2 x\sin^2 x+\sin^4 x$

from the last we can obtain $\sin^4x$ using that

$\cos^2 x=1-\sin^2x$
$\cos 2x=\cos^2x-\sin^2x=1-2\sin^2x$

edited Mar 11 '18 at 07:36

answered Mar 11 '18 at 07:02

user

154,566

1

I don't think that's true because (isinx)^4 = sin^4(x) which is Real – Varun Jahagirdar Mar 11 '18 at 07:05
Yes sorry, I indicated incorrectly at first, now I hope the method is clear. – user Mar 11 '18 at 07:30

Prove $ \sin^4(x)$ identity using $z=cis(x)?$

1 Answers1