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Answer 1 · 2018-02-11T14:57:32

Kindly refer to a Proof in the Explanation.

#tanx/(1-cotx)+cotx/(1-tanx)#,

#=tanx/(1-1/tanx)+(1/tanx)/(1-tanx)#,

#=tan^2x/(tanx-1)-1/{tanx(tanx-1)}#,

#=(tan^3x-1)/{tanx(tanx-1)}#,

#={cancel((tanx-1))(tan^2x+tanx+1)}/{tanxcancel((tanx-1))}#,

#=tan^2x/tanx+tanx/tanx+1/tanx#,

#=tanx+1+1/tanx#,

#=1+{sinx/cosx+cosx/sinx}#,

#=1+(sin^2x+cos^2x)/(sinxcosx)#,

#=1+1/(sinxcosx)#,

#=1+1/cosx*1/sinx#,

#=1+secxcscx,# as desired!