How do you prove #((1+cosx) / sinx) + (sinx / (1 + cosx)) = 2 csc x#?

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Answer 1 · 2015-04-15T21:42:42

#LHS#

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

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

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

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

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

#=2/sinx#

#=2cscx#

#=RHS#

This is because:

#a/b+c/d = (ad+bc)/(bd)#

And also because:

#cos^2x+sin^2x=1#