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

2 Answers
Mar 1, 2016

#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#

Mar 1, 2016

#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 #