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

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Answer 1 · 2016-08-30T18:35:02

The expression can be simplified to #2cscx#

Start by putting on a common denominator.

#=>((1 + cosx)(1 + cosx))/((sinx)(1 + cosx)) + (sinx(sinx))/((sinx)(1 + cosx))#

#=>(cos^2x + 2cosx + 1 + sin^2x)/(sinx(1 + cosx))#

Apply the pythagorean identity #cos^2x + sin^2x = 1#:

#=>(1 + 2cosx + 1)/(sinx(1 + cosx)#

#=>(2 + 2cosx)/(sinx(1 + cosx))#

Factor out a #2# in the numerator.

#=> (2(1 + cosx))/(sinx(1 + cosx))#

#=>2/sinx#

Finally, apply the reciprocal identity #1/sintheta = csctheta# to get:

#=> 2cscx#

Hopefully this helps!