# How do you solve tan^2x-2sec x+1=0 for 0°<=x<=360° ?

Sep 29, 2015

$x = {60}^{\circ} , x = {300}^{\circ}$

#### Explanation:

As ${\tan}^{2} \left(x\right) = {\sec}^{2} \left(x\right) - 1$ we can replace the ${\tan}^{2} \left(x\right)$ in the initial equation.

This leaves us with;

${\sec}^{2} \left(x\right) - 1 - 2 \sec \left(x\right) + 1 = 0$

After combining like terms the resulting formulae is:

${\sec}^{2} \left(x\right) - 2 \sec \left(x\right) = 0$

We can then factorise by $\sec \left(x\right)$ to yield;

$\sec \left(x\right) \left(\sec \left(x\right) - 2\right) = 0$

By the null factor law we can then establish two possible solutions for x: when $\sec \left(x\right) = 0$ or when $\sec \left(x\right) = 2$

As the secant of x can never equal zero this possibility can be discounted which leaves $\sec \left(x\right) = 2$.

To solve for $\sec \left(x\right) = 2$ first we take the reciprocal of each side;

$\cos \left(x\right) = \frac{1}{2}$

Now the values of x that satisfy this equality that are within the domain $\left[{0}^{\circ} , {360}^{\circ}\right]$ are; $x = {60}^{\circ} , x = {300}^{\circ}$.

Hope this helps :)