t,
$VP=(1-\sin x+\cos x)^2$
$=(1-\sin x)^2+2\cos x(1-\sin x)+\cos^2x$
$=1-2\sin x+\sin^2x+2\cos x-2\sin x\cos x+\cos^2x$
$=2\cos x-2\sin x-2\sin x\cos x+2$
$=2(\cos x-\sin x\cos x-\sin x+1)$
$=2[\cos x(1-\sin x)+(1-\sin x)]$
$=2(1-\sin x)(1+\cos x)$
$=VT$ (đpcm)
u,
$VT=\sin^3x(1+\cot x)+\cos^3x(1+\tan x)$
$=\sin^3x+\sin^3x.\dfrac{\cos x}{\sin x}+\cos^3x+\cos^3x.\dfrac{\sin x}{\cos x}$
$=\sin^3x+\cos^3x+\sin^2x\cos x+\sin x\cos^2x$
$=(\sin x+\cos x)(\sin^2x+\cos^2x-\sin x\cos x)+\sin x\cos x(\sin x+\cos x)$
$=(\sin x+\cos x)(1-\sin x\cos x+\sin x\cos x)$
$=\sin x+\cos x$
$=VP$ (đpcm)
v,
$VT=\sin^2x\tan x+\cos^2x\cot x+2\sin x\cos x$
$=\dfrac{\sin^3x}{\cos x}+\dfrac{\cos^3x}{\sin x}+2\sin x\cos x$
$=\dfrac{\sin^4x+\cos^4x}{\sin x\cos x}+2\sin x\cos x$
$=\dfrac{\sin^4x+\cos^4x+2\sin x\cos x.\sin x\cos x}{\sin x\cos x}$
$=\dfrac{(\sin^2x+\cos^2x)^2}{\sin x\cos x}$
$=\dfrac{1}{\sin x\cos x}$
$=\dfrac{\sin^2x+\cos^2x}{\sin x\cos x}$
$=\dfrac{\sin^2x}{\sin x\cos x}+\dfrac{\cos^2x}{\sin x\cos x}$
$=\dfrac{\sin x}{\cos x}+\dfrac{\cos x}{\sin x}$
$=\tan x+\cot x$
$=VP$ (đpcm)
