Question

Which of the following would be an acceptable first step in simplifying the expression sin(x)/1-cos(x)?

A. sin(x) - (sin(x)/cos(x))

B. 1/(csc(x))-(cos(x))

C. Cannot be simplified further

D. (sin(x)(1+cos(x))/(1-cos(x))(1+cos(x))

CORRECT ANSWER IS NOT C

Answer 1

D)
`(sin x)/(1-cos x) X(1+cos x)/(1+cosx)`

Explanation:

Given Expression

`(sin x)/(1-cos x)`

Rationalizing

=
`(sin x)/(1-cos x) X(1+cos x)/(1+cosx)`

=
`(sin x(1-cos x))/(1-cos^(2)x) }`

=
`(sin x(1-cos x))/(sin^(2)x) }`

cancellation, sin x we get

`= (1-cos x)/(sin x)\\ = \frac{2sin^(2) ({(x)/(2) } )}{2sin((x)/(2) ) cos((x)/(2)}` ( by using trigonometry formulas

`1-cos A = 2 sin^(2) ((A)/(2) )`

`sin A = 2 sin((A)/(2) ) cos((A)/(2) )` )

`= \frac{sin({(x)/(2) } )}{ cos((x)/(2)}`

`= tan ((x)/(2) )`

Final answer:-

`(sin x)/(1-cos x) = tan((x)/(2) )`