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[adelinabb]

a) \(\displaystyle{ \sin x+ \sin\left(x+ \frac{2 \pi}{3}\right)+ \sin\left(x+ \frac{4 \pi}{3}\right)=0}\)

b) \(\displaystyle{ \sin ^{6}x+ \cos ^{6}x+3 \sin ^{2} x \cos ^{2}x=1}\)

[RyHoO16]

b) \(\displaystyle{ \sin ^{6}x+ \cos ^{6}x+3 \sin ^{2} x \cos ^{2}x=1}\)

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

[anna_]

a) \(\displaystyle{ L=\sin x+ \sin\left(x+ \frac{2 \pi}{3}\right)+ \sin\left(x+ \frac{4 \pi}{3}\right)=sinx+2sin \frac{x+ \frac{2\pi}{3}+x+ \frac{4\pi}{3}}{2} cos \frac{x+ \frac{2\pi}{3}-x- \frac{4\pi}{3} }{2}=sinx+2sin \frac{2x+2\pi}{2}cos \frac{ \frac{-2\pi}{3} }{2} = sinx+2sin(x+\pi)cos(- \frac{\pi}{3})=sinx+2sin(x+\pi) (\frac{1}{2} )=sinx+sin(x+\pi)= 2sin \frac{x+x+\pi}{2} cos \frac{x-x-\pi}{2}= 2sin(x+ \frac{\pi}{2})cos(- \frac{\pi}{2}) =0=P}\)

[adelinab]

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