Can anyone help prove this trigonometric identity?

#sinx/(1+cos x) + cos x/sin x = csc x #

2 Answers
Nov 29, 2016

Cross multiply and simplify LHS (as below)

Explanation:

To prove: #sinx/(1+cosx) + cosx/sinx = cscx#

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

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

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

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

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

#=1/sinx = cscx = RHS#

Nov 29, 2016

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

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

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

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

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

# = 1/sinx#

# = cscx#