Euler’s formula help

[vast badge]

Hi, my textbook gives the expression in the picture on the right as the answer to the question on the left, but I don’t understand why you can’t just use Euler’s formula and get cos(4x) + isin(4x)

(I’m using x because I don’t know how to type theta)

[arctic carbon]

are you sure these two photos are the same question

[neat elk]

The question on the left doesn't have an expression as an answer.

[cosmic kite]

The image on the right is the expansion of $(\cos{x}+i\sin{x})^4$ using

the binomial Theorem. It is the same as $\cos{4x} + i\sin{4x}$ using

De Moivre's Theorem, as you have noted. Equation the real and

imaginary parts then leads to two trig identities:

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

and

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

[ancient valve]

Jay

[cosmic kite]

The image on the left is the equivalent result from the real parts of

$(\cos\theta + i\sin\theta)^3 = \cos{3\theta} + i\sin{3\theta}$

and, as it a result that you are to prove, no answer is needed.