Giải thích các bước giải:
1.Ta có:
$(1+\cot x)\sin^3x+(1+\tan x)\cos^3x$
$=(1+\dfrac{\cos x}{\sin x})\sin^3x+(1+\dfrac{\sin x}{\cos x})\cos^3x$
$=\sin^3x+\sin^2x\cos x+\cos^3x+\sin x\cos^2x$
$=\sin^3x+\cos^3x+\sin^2x\cos x+\sin x\cos^2x$
$=(\sin x+\cos x)(\sin^2x-\sin x\cos x+\cos^2x)+\sin x\cos x(\sin x+\cos x)$
$=(\sin x+\cos x)(\sin^2x+\cos^2x-\sin x\cos x)+\sin x\cos x(\sin x+\cos x)$
$=(\sin x+\cos x)(1-\sin x\cos x)+\sin x\cos x(\sin x+\cos x)$
$=(\sin x+\cos x)-\sin x\cos x(\sin x+\cos x)+\sin x\cos x(\sin x+\cos x)$
$=\sin x+\cos x$
2.Ta có:
$\dfrac{\sin5x}{\sin x}-\dfrac{\cos5x}{\cos x}$
$=\dfrac{\sin5x\cos x-\cos5x\sin x}{\sin x\cos x}$
$=\dfrac{\sin(5x-x)}{\sin x\cos x}$
$=\dfrac{\sin4x}{2\sin x\cos x}$
$=\dfrac{2\sin2x\cos2x}{\dfrac12\sin2x}$
$=4\cos2x$
