Question

докажите тождества
1) cos2t/(sintcost+sin^2t)=ctg(pi+t)-1
2) (sin2t-2sin(pi/2-t))/(cos(pi/2-t)-sin^2t=-2ctgt
3)(ctgt-tgt)sin2t=2cos2t
срочно надо)

Answer 1

3) ((cosx*sin2x)/(sinx))-((sinx*sin2x)/(cosx))=((cosx*2sinx*cosx)/(sinx))-((sinx*2sinx*cosx)/(cosx))=2cos^2(x)-2sin^2(x)=2*(cos^2(x)-sin^2(x))=2*cos2x

Answer 2

(cos(2x))/(sinx*cosx+sin^2(x))=(cos^2(x)-sin^2(x))/(sinx*(cosx+sinx))=(sinx^2(x)*(ctg^2(x)-1))/(sinx*(cosx+sinx))=(sinx*(ctg^2(x)-1))/(cosx+sinx)=(sinx*(ctg^2(x)-1))/(sinx*(ctgx+1))=(ctg^2(x)-1)/(ctgx+1)=((ctgx-1)*(ctgx+1))/(ctgx+1)=ctgx-1=ctg(pi+t)-1

Answer 3

// sin(pi/2-x) = cosx
// cos(pi/2-x) = sinx
// (sinx-1)/(1-sinx) = -1

(sinx2x-2cosx)/(sinx-sin^2(x)) = (2sinx*cosx-2cosx)/(sinx-sin^2(x))=(2cosx*(sinx-1))/(sinx*(1-sinx))=-2ctgx