Simplify trig expression?

Question

Rewrite sin(x)/(1-sec(x)) without a fraction. Appraetnly answer is -cot(x)(cos(x)+1), but I don't know how to get there.

Answer 1

sin(x)/(1 - sec(x))

= -cos(x) cot(x/2)

Answer 2

sin(x)/(1-sec(x))

multiply top and bottom by 1+sec(x)

= sin(x)(1+sec(x))/(1-sec²(x))

multiply top and bottom by cos²(x) : recall cos(x)sec(x) = 1

= sin(x)cos(x)(cos(x)+1)/(cos²(x) - 1)

apply the Pythagorean Identity to the denominator

= sin(x)cos(x)(cos(x) + 1)/(-sin²x)

divide top and bottom by -sin(x)

= -cos(x)(cos(x)+1)/sin(x)

apply the identity cos(x)/sin(x) = cot(x)

= -cot(x)(cos(x) + 1)

QED

Answer 3

Multiply both top and bottom by (1+sec(x))

The denominator reduces to (1²-sec²(x))

Then multiply both top and bottom by cos²(x)

The denominator then reduces to cos²(x)-1 which equals (-sin²(x))

I'll let you do the rest for yourself.