# using identity to solve for x

• Nov 25th 2007, 01:20 PM
kenan
using identity to solve for x
the question says: Use an identity to find solutions on the interval [0,2pi]

sec^2 x + tanx = 3

thanks
• Nov 25th 2007, 02:18 PM
Soroban
Hello, kenan!

Quote:

Use an identity to find solutions on the interval $\displaystyle [0,\,2\pi]$

. . $\displaystyle \sec^2\!x + \tan x \:= \:3$

Use: .$\displaystyle \sec^2\!\theta \;=\;\tan^2\!\theta + 1$

Then we have: .$\displaystyle \tan^2\!x + 1 + \tan x \;=\;3$

. . which is a quadratic:] .$\displaystyle \tan^2\!x + \tan x - 2 \:=\:0$

. . which factors: .$\displaystyle (\tan x -1)(\tan x + 2)\:=\:0$

Then we have:
$\displaystyle \tan x -1 \:=\:0\quad\Rightarrow\quad \tan x \:=\:1\quad\Rightarrow\quad x \:=\:\frac{\pi}{4},\:\frac{5\pi}{4}$

$\displaystyle \tan x + 2 \:=\:0\quad\Rightarrow\quad \tan x \:=\:-2\quad\Rightarrow\quad x \:=\:\tan^{-1}(-2) \:\approx\:2.034,\;5.176$

• Nov 25th 2007, 02:24 PM
kenan
hey thanks for the answer:), but can you show me how you derived
$\displaystyle \sec^2\!\theta \;=\;\tan^2\!\theta + 1$

ive never seen that identity before so I'm just curious where it comes from