1.
а)
х²/(х²-у²) * (х-у)/х = х²/(х-у)(х+у) * (х-у)/х = х/(х+у),
б)
а/(3а+3в) : а²/(а²-в²) = а/(3*(а+в)) : а²/(а-в)(а+в) =
= а/(3*(а+в)) * (а-в)(а+в)/а² = (а-в)/3а,
в)
(-2с³/у)⁵ = -32с¹⁵/у⁵
г)
х/у² * 4ху = 4х²/у
2.
( у/(у-х) - (у-х)/у ) * (у-х)/х =
= ( у² - (у-х)²) / (у-х)у ) * (у-х)/х =
= ( у²-у²+2ху-х² ) / (у-х)у ) * (у-х)/х =
= х(2у-х) / (у-х)у ) * (у-х)/х = (2у-х) / у,
3.
(2х-4)/(х²+12х+36) : (8х-16)/(х²-36) =
= 2*(х-2)/(х+6)² : 8*(х-2)/(х-6)(х+6) =
= 2*(х-2)/(х+6)² : (х-6)(х+6)/8*(х-2) =
= (х-6) / 2*(х+6),
при х = 1,5:
(1,5-6) / 2*(1,5+6) = -4,5 / (2*7,5) = -4,5 / 15 = -3/10 (или -0,3)
4.
( а-8 + 32а/(а-8) ) * ( 8+а - 32а/(8+а) ) =
= [ ( (а-8)²+32а )/(а - 8) ] * [ ( (8+а)²-32а)/(8+а) ] =
= (а²-16а+64+32а)/(а-8) * (64+16а+а²-32а)/(8+а) =
= (а²+16а+64)/(а-8) * (а²-16а+64)/(8+а) =
= (а+8)²/(а-8) * (а-8)²/(8+а) =
= (а + 8)(а - 8) = а² - 64
рукописный вариант:
⇅⇅⇅⇅
sinx+sin5x-√2 sin3x=0
2sin3xcos2x-√2sin3x=0
sin3x(2cos2x-√2)=0
sin3x=0⇒3x=πn⇒x=πn/3,n∈Z
cos2x=√2/2⇒2x=+-π/4+2πn⇒x=+-π/8+πn,n∈Z
2)cos(70º+x)cos(x-20º)=1/2
1/2(cos90+cos(2x+50))=1/2
cos(2x+50)=1⇒2x+50=360n⇒2x=-50+360n⇒x=-25+180n,n∈Z
3)sin3x-√3 cos2x=sinx
sin3x-sinx-√3 cos2x=0
2sinxcos2x-√3cosx=0
cos2x(2sinx-√3)=0
cos2x=0⇒2x=π/2+πn⇒x=π/4+πn,n∈Z
sinx=√3/2⇒x=(-1)^n*π/3+πn,n∈Z
4)4cos^2x+sinxcosx+3sin^2x-3=0
4cos^2x+sinxcosx+3sin^2x-3sin²x-3cos²x=0
cos²x+sinxcosx=0/cos²x≠0
tgx+1=0
tgx=-1⇒x=-π/4+πn,n∈Z
5)cos^2x-3sinxcosx=-1
cos^2x-3sinxcosx+sin²x+cos²x=0/cos²x≠0
tg²x-3tgx+2=0
tgx=a
a²-3a+2=0
a1+a2=3 U a1*a2=2
a1=1⇒tgx=1⇒x=π/4+πn,n∈Z
a2=2⇒tgx=2⇒x=arctg2+πn,n∈Z