Expand cos4A?

Well,

first of all, welcome to this forum from (maybe the only) french guy in the zone !! ;-)

edit : fooks (AUTOPROMOTED) answer is unfortunately a little ridiculous :

THE GOAL consists in writing cos(4a) with exclusive powers of cosx
IOW,
to write :
cos(4x) = P( cosx) where P is a POLYNOMIAL !!

this type of exercise is calle LINEARISATION

PARFOIS, ils sont vraiment trop nuls, ces anglo-saxons !! ;-)


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using De_Moivre's formula, we can get cos(4x) and sin(4x) simultaneously !!

cos(4x) + i sin(4x) = [ cosx + i sinx ]^4 <--- use binomial formula

and you will also need cos(3x) and sin(3x), so you need to do :
cos(3x) + i sin(3x) = [ cosx + i sinx ]^3
...
now, as I did the exercise yesterday, you are luckys : here it is :
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cos(3x) and (sin(3x) :

[ cosx + i sinx ]^3 = cos(3x) + isin(3x)
therefore :
expand the LHS by using (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
group real part and imaginary part and finally : identify with RHS
can you do this ?

[ cosx + i sinx ]^3 = cos^3x + 3icos^2x sinx - 3cosx sin^2x -isin^3x
= cos^3x - 3cosx(1 - cos^2x) + i*[ - sin^3x + 3sinx*(1 - sin^2x) ]
= 4cos^3x - 3cosx + i*[ - 4sin^3x + 3sinx]
conclusion :
as two complex numbers are equal iif both, their real part and imaginary part are equal
we obtain :

cos(3x) = 4cos^3x - 3cosx
and
sin(3x) = - 4sin^3x + 3sinx
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proceed the same way with degree/power = 4
binomial formula is :
(a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4

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alternative and more practical method :

cos(4x) = cos(2x + 2x) = cos(2x)cos(2x) - sin(2x)sin(2x)
= (2cos^2 - 1)^2 + sin(2x)^2
= (2cos^2 - 1)^2 + 4sin^2xcos^2x
= (2cos^2 - 1)^2 + 4cos^2x (1 - cos^2x)
... and you have a polynomial in cosx !!

et voilà !!

hope it' ll help !!

PS: if you want good answers, please do not forget to give Best Answers to one of those who answer.....
to me, or anybody else, or course provided the answer deserves one !
this is to encourage people to answer !! ;-)

cos(4A)

= cos 2(2A)

= cos²(2A) - sin²(2A)

= (cos²A - sin²A)² - (2sinA cosA)²

= [cos⁴(A) - 2 sin²(A) cos²(A) + sin⁴(A)] - 4 sin²(A) cos²(A)

= cos⁴(A) - 6 sin²(A) cos²(A) + sin⁴(A)