Simplify the following trig expression?

Question

(1-sinx)(1-cscx)sinx

Answer 1

The expression is (1-sinx)(1-cosecx)sinx

=  [ (1-cosecx) -sinx ( 1-cosecx) ]sinx

=  [ (1-cosecx) -sinx +sinx cosecx ] sinx        ( cosecx = 1/ sinx )

= [ 1- cosecx -sinx +(sinx . 1/ sinx) ] sinx

= [ 1-cosecx -sinx +1 ] sinx

= [ sinx - sinx cosecx - sin²x +sinx

=  sinx - (sinx .1/ sinx) -sin²x +sinx

= sinx -1- sin²x +sinx

=  -(sin²x -2sinx +1)

Therefore the solution is  -( sinx - 1)².