Question

How do you prove `(1+tanx)/(1+cotx)=2`?

Answer 1

This identity is false!!!

Explanation:

Simplifying the left side:

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

`((cosx + sinx)/cosx)/((sinx + cosx)/sinx)`

`(cosx + sinx)/cosx xx sinx/(sinx + cosx) =`

`sinx/cosx`

`tanx`

Hopefully this helps!

Answer 2

Another way to prove this false is as follows.

Since `(tanx)/(tanx) = 1` and `tanxcotx = tanx*1/(tanx) = 1`:

`color(blue)((1+tanx)/(1+cotx))*(tanx)/(tanx)`

`= (tanx(1+tanx))/(tanx + tanxcotx)`

`= (tanxcancel((1+tanx)))/cancel(1+tanx)`

`= color(blue)(tanx)`