## anonymous 3 years ago find all solutions of the equation in the terminal [0, 2 pi). sec^2 x + tan x = 1

1. anonymous

2. anonymous

Use the fact that $sec^{2} x = \tan^{2}x + 1$ and therefore the equation becomes, after substitution and some algebraic manipulation, $\tan^{2}x +\tan x = 0$ Then factor to get $\tan x(\tan x +1)=0$ and therefore either $tan x = 0$ or $\tan x + 1=0$ In the first case, then on this interval, either $x = 0$ or $x = \pi$ In the second case, then we get $\tan x = -1$ and therefore $x=\frac{3\pi}{4}$ or $x=\frac{7\pi}{4}$.

3. anonymous

Well: $\sec^2(x)=1+\tan^2(x) \implies \tan^2(x)+\tan(x)+1=1$ $\implies \tan(x)(\tan(x)+1)=0 \implies \tan(x)=0, \tan(x)=-1; x=n \pi, x=\frac{(4n+3) \pi}{4}$

4. anonymous

The last part should be: $\frac{(4n+3) \pi}{4}$

5. anonymous

thank you