Question

Упростите выражение:
1) $\frac{sin ^{4} \alpha -cos^4 \alpha }{cos^2 \alpha -sin^2 \alpha } -tg^2 \alpha ctg^2 \alpha$
2)$cos( \pi +2 \alpha )+sin( \pi +2 \alpha )tg( \frac{ \pi }{2}+ \alpha )$
3)$\frac{sin^3 \alpha -cos^3 \alpha }{1+sin \alpha cos \alpha } +cos \alpha -sin \alpha$

Answer 1

1. (sin⁴α-cos⁴α)/(cos²α-sin²α))-tg²α*ctg²α=-2
sin⁴α-cos⁴α=(sin²α-cos²α)*(sin²α+cos²α)=(sin²α-cos²α)*1=sin²α-cos²α
(sin²α-cos²α)/(cos²α-sin²α)=-1
tg²α*ctg²α=(tgα*ctgα)²=1²=1
-1-1=-2

2. cos(π+2α)+sin(π+2α)*tg(π/2+α)=-cos2α-sin2α*(-ctgα)=-cos2α+sin2α*(cosα/sinα)=-(cos²α-sin²α)+2sinα*cosα*cosα/sinα=-cos²α+sin²α+2cos²α=sin²α+cos²α=1

3. (sin³α-cos³α)/(1+sinα*cosα)+cosα-sinα=[(sinα-cosα)*(sin²α+sinα*cosα+cos²α)] / (1+sinα*cosα)+cosα-sinα=[(sinα-cosα)*(1+sinα*cosα)] / (1+sinα*cosα)+cosα-sinα=sinα-cosα+cosα-sinα=0