Question

cos^3α/sin^3α

Answer 1

问：cos^3α/sin^3α

答：sin^3α+cos^3α=1（sinα+cosα)(sin^2α-sinαcosα+cos^2α)=1(sinα+cosα)(1-sinαcosα)=1令x=sinα+cosαx^2=sin^2α+2sinαcosα+cos^2α=1+2sinαcosα所以sinαcosα=(x^2-1)/2所以x[1-(x^2-1)/2]=12x-2x^3+x=2x^3-3x+2=0x^3-x-2x+2=0x(x+1)(x-1)-2(x-1)=0(x-1)(x²+x-2)=0(x-1)^2(x+2)=0x=1,x=-2x=sinα+cosα=√2sin(α+π/4)所以x>=-√2所以x=1sinα+cosα=1