proving Trig Identitity: (1- tanx)^2 = sec^2x - 2tanx
- Thread starter Hockeyman
- Start date
Hello, Hockeyman!
Going the other way . . .
We have: \(\displaystyle \
1\,-\,\tan x)^2\;=\;1\,-\,2\cdot\tan x\,+\,\tan^2x \;=\;\underbrace{1\,+\,\tan^2x}\,-\,2\cdot\tan x\)
. . . . . . . . . Since \(\displaystyle 1\,+\,\tan^2x\:=\:\sec^2x\), we have:. . . \(\displaystyle \sec^2x\,-\,2\cdot\tan x\)
Going the other way . . .
We have: \(\displaystyle \
1\,-\,\tan x)^2\;=\;1\,-\,2\cdot\tan x\,+\,\tan^2x \;=\;\underbrace{1\,+\,\tan^2x}\,-\,2\cdot\tan x\). . . . . . . . . Since \(\displaystyle 1\,+\,\tan^2x\:=\:\sec^2x\), we have:. . . \(\displaystyle \sec^2x\,-\,2\cdot\tan x\)