Question

how to verify this identity (cosxcotx)/(1-sinx)-1=cscx?

Answer 1

cos(x)cot(x)/(1-sin(x))=cos^2(x)/(sin(x)(1-sin(x)). Multiply top and bottom by 1+sin(x):

cos^2(x)(1+sin(x))/(sin(x)(1-sin^2(x))=cos^2(x)(1+sin(x))/(sin(x)cos^2(x))=(1+sin(x))/sin(x)=cosec(x)+1.

So cos(x)cot(x)/(1-sin(x))-1=cosec(x)

Answer 2

cot x = cosx/sinx

cosx(cosx/sinx)/1-sinx - 1 = cscx

cos^2x/sinx(1-sinx) - 1 = cscx

cos^2x/sinx(1-sinx) - sinx(1-sinx)/sinx(1-sinx) = cscx

[cos^2x - sinx(1-sinx)] / sinx(1-sinx) = csc x

[cos^2x - sinx + sin^2x]/ sinx(1-sinx) = csc x

but cos^2x + sin^2x = 1

1 - sinx/sinx(1-sinx) = cscx

1 - sinx cancels each other we have;

1/sinx =cscx

and cosec x is equal to 1 / sine x trigonometrically