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

2 Answers
sente
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

Jim G.
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

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