How do you verify #(tan(x)/(1 + sec(x))) + (1+sec(x)/tan(x)) = 2csc(x)#?

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Answer 1 · 2016-04-16T22:37:23

See below

LHS=left hand side, RHS=right hand side

LHS#=(tanx)/(1+secx) +(1+secx)/tanx #

#=((tanx)(tanx)+(1+secx)(1+secx))/(tanx(1+secx))#

#=(tan^2x+1+2secx+sec^2x)/(tanx(1+secx))#

#=((tan^2x+1)+2secx+sec^2x)/(tanx(1+secx))#

#=(sec^2x +2secx+sec^2x)/(tanx (1+secx))#

#=(2sec^2x+2secx)/(tanx(1+secx))#

#=(2secx(secx+1))/(tanx(1+secx))#

#=2 secx/tanx#

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

#=2 1/cosx xx cosx/sinx#

#=2 1/sinx#

#=2cscx#

#=RHS#