#tanx/(1+secx)+(1+secx)/tanx!=2secx#. It should rather be #tanx/(1+secx)+(1+secx)/tanx=2cscx#
Perhaps what you mean is to prove that #tanx/(1+secx)+(1+secx)/tanx=2cscx#. To solve this let us start from LHS and prove using other identities that this is equal to RHS.
#tanx/(1+secx)+(1+secx)/tanx# (let us first add them like fractions)
= #(tan^2x+(1+secx)^2)/(tanx(1+secx)# (expanding this)
= #(tan^2x+1+sec^2x+2secx)/(tanx(1+secx)#(using #sec^2x=1+tan^2x#)
= #(sec^2x+sec^2x+2secx)/(tanx(1+secx)# or
=#(2sec^2x+2secx)/(tanx(1+secx)# or
= #(2secx(secx+1))/(tanx(1+secx)#
Now, using #secx=1/cosx# and #tanx=sinx/cosx#
= #2secx/tanx# =#(2/cosx)/(sinx/cosx)# (simplifying & using #1/sinx=cscx#)
= #(2/sinx)# = #2cscx#
