Verify that - tan2x = (2tanx) / (sec^2x - 2)

Below.

Step-by-step explanation:

- tan 2x = - 2tanx / (1 - tan^2x)

Using the identity tan^2x = sec^2x - 1 and substituting for tan^2x:

- tan 2x = - 2 tanx / (1 - (sec^2x - 1))

= 2 tanx / ( - 1 (sec^2x + 2))

= 2 tan x / (sec^2 x - 2)