simplify tanx sinx(cos^2x cscx - sinx)

Question

Answer 1

It might be useful to turn everything into sines and cosines.

tan(x) = sin(x)/cos(x)

csc(x) = 1/sin(x)

So the problem becomes:

Answer 2

have tant(x)(cos²(x)-sin²(x))=tan(x)cos(2x)

Answer 3

tan(x)sin(x)[cos²(x)csc(x)-sin(x)] = tan(x)cos²(x)-sin²(x)tan(x) = sin(x)cos(x)-sin²(x)tan(x)

Because sin(x) and csc(x) are reciprocals of each other, they get cancelled out as a result when distributing! Also, by the quotient identity tan(x)=sin(x)/cos(x), then cos(x) in the denominator and cos²(x) are cancelled out and we're left with sin(x)cos(x) for the resultant's first term.