`~rai~`

\(4.(1+\tan x)\cos^2x+(1+\cot x)\sin^2x=(\sin x+\cos x)^2\\\text{Xét VT=}(1+\tan x)\cos^2x+(1+\cot x)\sin^2x\\\quad\quad\quad=\cos^2x+\dfrac{\sin}{\cos x}.\cos^2x+\sin^2x+\dfrac{\cos x}{\sin x}.\sin^2x\\\quad\quad\quad=\cos^2x+\sin x\cos x+\sin^2x+\sin x\cos x\\\quad\quad\quad=\sin^2x+2\sin x\cos x+\cos^2x\\\quad\quad\quad=(\sin x+\cos x)^2\\\quad\quad\quad=VP.\\\Rightarrow (1+\tan x)\cos^2x+(1+\cot x)\sin^2x=(\sin x+\cos x)^2.(đpcm)\\5.\sin^2x\tan x+\cos^2x\cot x+2\sin x\cos x=\tan x+\cot x\\\text{Xét VT=}\sin^2\tan x+\cos^2\cot x+2\sin x\cos x\\\quad\quad\quad=\sin^2x.\dfrac{\sin x}{\cos x}+\cos^2x.\dfrac{\cos x}{\sin x}+2\sin x\cos x\\\quad\quad\quad=\dfrac{\sin^3x}{\cos x}+\dfrac{\cos^3x}{\sin x}+2\sin x\cos x\\\quad\quad\quad=\dfrac{\sin^4x}{\sin x\cos x}+\dfrac{\cos^4x}{\sin x\cos x}+\dfrac{2\sin^2x\cos^2x}{\sin x\cos x}\\\quad\quad\quad=\dfrac{\sin^4x+\cos^4x+2\sin^2x\cos^2x}{\sin x\cos x}\\\quad\quad\quad=\dfrac{(\sin^2x+\cos^2x)^2-2\sin^2x\cos^2x+2\sin^2x\cos^2x}{\sin x\cos x}\\\quad\quad\quad=\dfrac{1}{\sin x\cos x}\\\quad\quad\quad=\dfrac{\sin^2x+\cos^2x}{\sin x\cos x}\\\quad\quad\quad=\dfrac{\sin^2x}{\sin x\cos x}+\dfrac{\cos^2x}{\sin x\cos x}\\\quad\quad\quad=\dfrac{\sin x}{\cos x}+\dfrac{\cos x}{\sin x}\\\quad\quad\quad=\tan x+\cot x\\\quad\quad\quad=VP.\\\Rightarrow \sin^2x\tan x+\cos^2x\cot x+2\sin x\cos x=\tan x+\cot x.(đpcm)\)

`4, (1+tanx).cos^2x+(1+cotx).sin^2x`

`=(1+(sinx)/(cosx)).cos^2x+(1+(cosx)/(sinx)).sin^2x`

`=cos^2x+sinx.cosx+sin^2x+cosx.sinx`

`=sin^2x+2sinx.cosx+cos^2x`

`=(sinx+cosx)^2 (\text{đpcm})`

`5, sin^2x.tanx+cos^2x.cotx+2sinx.cosx`

`=sin^2x. (sinx)/(cosx)+cos^2x.(cosx)/(sinx)+2sinx.cosx`

`=(sin^4x+cos^4x+2sin^2x.cos^2x)/(sinx.cosx)`

`=(sin^2x+cos^2x)^2/(sinx.cosx)`

`=1/(sinx.cosx)`

`=(sin^2x+cos^2x)/(sinx.cosx)`

`=(sin^2x)/(sinx.cosx)+(cos^2x)/(sinx.cosx)`

`=(sinx)/(cosx)+(cosx)/(sinx)`

`=tanx+cotx (\text{đpcm})`