Question

How do you verify the identity `(csc x - sin x)(sec x - cos x)(tan x + cot x) = 0`?

Answer 1

I am not sure that is equal to zero....I got `1`!

Have a look.

Answer 2

WARNING: This may not be the simplest way!
`(csc - sin)(sec - cos)(tan+cot)`
Let `s` represent `sin(x)`
`c` represent `cos(x)`
and `t` represent `tan(x)`

`(csc - sin)(sec - cos)(tan+cot)`
becomes
`(1/s - s)(1/c-c)(t+1/t)`

`=(1/(sc)-t - 1/t+sc)*(t+1/t)`

`((xx,|1/(sc),,-t,,-1/t,,+sc),( - , - , - , - , - , - , - , - - ),(1/t,|1/(sct),,-1,,-1/t^2,,+(sc)/t),(+t,|t/(sc),,-t^2,,-1,,sct) )`

Replacing `t` with `s/c`

we get
`1/(s^2) - 1 -c^2/s^2 +c^2 +1/c^2 -s^2/c^2 -1 +s^2`

`=(1-s^2)/(c^2) -2 +(c^2+s^2) +(1-c^2)/s^2`

`= 1`