Question

Find all solutions in the interval [0, 2π). sec2 x - 2 = tan2 x

Answer 1

The equation is equivalent to 1/cos(2x)-2=sin(2x)/cos(2x). Make the denominator cos(2x) for all terms, we get 1-2cos(2x)=sin(2x). Since cos^2(2x)+sin^2(2x)=1, let cos(2x)=a, substitute sin(2x) by +-\sqrt(1-a^2), and solve the equation with only one variable a. We get a=4/5 or 0, but cos(2x) cannot be 0, otherwise sec(2x) is undefined. 2x=36.9 degrees, so x=18.9 degrees approximately.