Question

Solve the trig equation by factoring. step by step please
sec^2(x)-sec(x)=2

Answer 1

`\sec^2(x)-\sec(x)=2\implies [\sec(x)]^2-\sec(x)=2 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{let's first just a few make}}{\sec(x)=Z} \\\\\\ Z^2-Z=2\implies Z^2-Z-2=0 \implies (Z+1)(Z-2)=0 \\\\[-0.35em] ~\dotfill`

`Z+1=0\implies \sec(x)+1=0\implies \sec(x)=-1\implies \cfrac{1}{\cos(x)}=-1 \\\\\\ \cfrac{1}{-1}=\cos(x)\implies -1=\cos(x)\implies \cos^(-1)(-1)=x\implies \boxed{\pi =x} \\\\[-0.35em] ~\dotfill\\\\ Z-2=0\implies \sec(x)-2=0\implies \sec(x)=2\implies \cfrac{1}{\cos(x)}=2 \\\\\\ \cfrac{1}{2}=\cos(x)\implies \cos^(-1)\left( \cfrac{1}{2} \right)=x\implies \boxed{\cfrac{\pi }{3}~~,~~\cfrac{5\pi }{3}=x}`