Question

How do you simplify `(sinx/(1-cosx))+((1-cosx)/sinx)`?

Answer 1

`sinx/(1-cosx)+(1-cosx)/sinx = 2cscx`

Explanation:

`sinx/(1-cosx)+(1-cosx)/sinx`

Multiply the first term by `sinx/sinx` and the second term by `(1-cosx)/(1-cosx)`

`= sin^2x/(sinx(1-cosx))+(1-cosx)^2/(sinx(1-cosx))`

Group terms with common denominators

`=(sin^2x+(1-cosx)^2)/(sinx(1-cosx))`

Expand `(1-cosx)^2`

`=(sin^2x+1-2cosx+cos^2x)/(sinx(1-cosx))`

Apply the identity `sin^2x+cos^2x=1`

`=(2-2cosx)/(sinx(1-cosx))`

Factor out `2` from the numerator

`=(2(1-cosx))/(sinx(1-cosx))`

Cancel common terms from the numerator and denominator

`=2/sinx`

Apply the definition of the cosecant function (`cscx = 1/sinx`)

`=2cscx`

Answer 2

`2/sinx`

Explanation:

Write with a common denominator

`(sin^2x + (1 - cosx)^2)/(sinx(1 - cosx))`

`=( sin^2x + 1 - 2cosx + cos^2x)/(sinx(1- cosx))`

`=( sin^2x + cos^2x + 1 - 2cosx)/(sinx(1-cosx))`

[using the identity : `sin^2x + cos^2x = 1]`

then becomes :`( (1 + 1 - 2cosx))/(sinx(1-cosx))`

`= (2(1 - cosx))/(sinx(1-cosx))`

`=( 2cancel(1-cosx))/(sinxcancel(1-cosx)) = 2/sinx`