prove that `cos^-1(sinx)= pi/2-x`

If |x|<=1 then prove that cos^(-1)(-x)=pi-cos^(-1)x

If cos^(-1)x+cos^(-1)y=pi/2 then prove that cos^(-1)x=sin^(-1)y

The area bounded by the curve y=|cos^-1(sinx)|+|pi/2-cos^-1(cosx)| and the x - axis , where pi/2lexlepi , is equal to

Prove that tan^(-1)((cosx-sinx)/(cosx+sinx))=(pi/4-x), x lt pi .

Prove that: cos((3 pi)/(2)+x)cos(2 pi+x){cot((3 pi)/(2)-x)+cot(2 pi+x)}=1

Prove that: cos((3 pi)/(2)+x)cos(2pi+x)[cot((3 pi)/(2)-x)+cot(2 pi+x)]=1

Prove that: (a) (cos(pi+x) cos(-x))/(sin(pi-x)cos(pi/2+x))=cot^(2)x (b) cos((3pi)/2 + x)cos(2pi+x){cot ((3pi)/2-x)+cot(2pi+x)}=1

Prove that cos x+cos((2 pi)/(3)-x)+cos((2 pi)/(3)+x)=0

prove that cos2(pi)/(7)+cos4(pi)/(7)+cos6(pi)/(7)=-(1)/(2)

Prove that: (cos(pi+x)cos(-x))/(sin(pi-x)cos(pi)/(2)+x)=cot^(2)x