[TEX]sin (x+\frac{\pi}{4}) sin^{3} 3x+ cos (3x + \frac{\pi}{4})cos ^{3}x = 0[/TEX]
[TEX](sin x +cos x)sin^{3} 3x+(cos 3x - sin 3x)cos ^{3}x = 0[/TEX]
[TEX]\frac{sin x + cos x}{cos ^{3} x}= \frac{cos(\pi - 3x) + sin (\pi-3x)}{ sin^{3}( \pi - 3x)}[/TEX]
[TEX]tan^{3} x + tan^{2} x + tan x+1= cot^{3}(\pi - 3x)+ cot^{2}(\pi-3x)+cot(\pi-3x)+1[/TEX] ( 1 )
ta xet hàm số
[TEX]f(m)= m^{3}+m^{2}+m+1 (m\in\mathbb{R})[/TEX]
[TEX]f^{'}(m)=3m^{2}+2m+1 >0 \forall m\in\mathbb{R}[/TEX]
suy ra f(m) đồng biến trên toàn R.
khi đó :
( 1 ) <=> [TEX] tan (x) = cot (\pi - 3x)[/TEX]
<=> [TEX] x = \frac{\pi}{4} + \frac {k \pi} {2}[/TEX]