using identity to solve for x

**kenan** · November 25th 2007, 01:20 PM

the question says: Use an identity to find solutions on the interval [0,2pi]

sec^2 x + tanx = 3

thanks

**Soroban** · November 25th 2007, 02:18 PM

Hello, kenan!

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

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

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

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

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

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

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

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

**kenan** · November 25th 2007, 02:24 PM

hey thanks for the answer

, but can you show me how you derived
$<br /> \sec^2\!\theta \;=\;\tan^2\!\theta + 1<br />$

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

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