Question

How do you prove `csc^2theta-cot^2theta=cotthetatantheta`?

Answer 1

Use the identities `cscbeta = 1/sinbeta`, `cotbeta = cosbeta/sinbeta` and `tanbeta = sin beta/cosbeta`.

`1/sin^2theta - cos^2theta/sin^2theta = costheta/sintheta * sintheta/costheta`

`(1 - cos^2theta)/sin^2theta = 1`

Use the identity `sin^2beta + cos^2beta = 1`.

`sin^2theta/sin^2theta = 1`

`1 = 1`

`LHS = RHS`

Hopefully this helps!

Answer 2

`csc^2theta-cot^2theta=cotthetatantheta`

Use the Pythagorean identity `1+cot^2theta=csc^2theta`

`1+cot^2theta -cot^2theta=cotthetatantheta`

`1=cotthetatantheta`

`tantheta/tantheta=cotthetatantheta`

`tantheta * 1/tantheta =cotthetatantheta`

Use the identity `cottheta=1/tantheta`

`tantheta * cottheta= cotthetatantheta`

`cotthetatantheta=cotthetatantheta`