For `0ltxltpi/ 2,` prove that `cos(sinx)gtsin(cosx)`

If 0ltxltpi/2 prove that sqrt(tanx+sinx)+sqrt(tanx-sinx)=2sqrt(tanx)cos (pi/4-x/2)

Prove that (1-sinx)/(cosx)=(cosx)/(1+sinx)

Prove that: (sinx+sin3x)/(cosx+cos3x)=tan2x

If x+y+z=pi/2, then prove that |[sinx,siny,sinz],[cosx,cosy,cosz],[cos^3x,cos^3 y,cos^3z]|=0

Prove that (cosx)/(1-sinx)=(1+cosx+sinx)/(1+cosx-sinx)

Prove that (sin5x-2sin3x+sinx)/(cos5x-cosx)=tanx

Prove that sin1>cos(sin1)dot Also, show that the equation sin("cos"(sinx))=cos("sin"(cosx)) has only one solution in [0,pi/2]dot

If y=(sinx)/(1+(cosx)/(1+(sinx)/(1+(cosx)/(1+\ ddotto\ oo)))) , prove that (dy)/(dx)=((1+y)cosx+ysinx)/(1+2y+cosx-sinx)

Prove that (sin-cosx+1)/(sinx+cosx-1)=secx+tanx

If y=sqrt(sinx+sqrt(sinx+sqrt(sinx+\ dotto\ oo))) , prove that (dy)/(dx)=(cosx)/(2y-1)