Key to "Hint to prove sin4x+cos4x=3+cos⁡(4x)4"

High School
Geometry
Trigonometry
Trigonometric equations and identities

PEEWSRIGWETRYqx

Answered question

2021-12-30

Hint to prove

\sin^{4} x + \cos^{4} x = \frac{3 + \cos (4 x)}{4}

Answer & Explanation

alexandrebaud43

Beginner2021-12-31Added 36 answers

\frac{3 + \cos (4 x)}{4} = \frac{3 + 2 \cos^{2} 2 x - 1}{4} = \frac{{(\cos^{2} x - \sin^{2} x)}^{2} + 1}{2}

= \frac{\sin^{4} x + \cos^{4} x + {(\sin^{2} x + \cos^{2} x)}^{2} - 2 \sin^{2} x \cos^{2} x}{2}

= \sin^{4} x + \cos^{4} x

kalfswors0m

Beginner2022-01-01Added 24 answers

$1 - 2 \sin^{2} x \cos^{2} x = 1 - (1 - \cos^{2} x) \cos^{2} x - \sin^{2} x (1 - \sin^{2} x)$
$= 1 - \cos^{2} x + \cos^{4} x - \sin^{2} x + \sin^{4} x$
$= \cos^{4} x + \sin^{4} x$

Vasquez

Expert2022-01-09Added 669 answers

Well, you can always use the Euler formulas
$\cos x = \frac{e^{i x} + e^{- i x}}{2}, \sin x = \frac{e^{i x} - e^{- i x}}{2 i}$
and expand the powers on the left side if you are ok with that solution.

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