sec^2 (x) -2 = tan^2 (x)
Rewritng sec and tan in terms of sin and cos
1/ cos ^2(x) - 2 = sin ^2(x) / cos^2(x)
Multiply each term by cos^2(x)
1 - 2 cos^2(x) = sin^2(x)
Add 2 cos^2(x) to each side
1 = sin^2(x) + 2cos^2(x)
Break 2cos^2(x) into cos^2 (x) + cos^(x)
1 = sin^2(x) + cos^2(x) + cos^2(x)
We know that sin^2(x) + cos^2(x) = 1
1 = 1+cos^2(x)
Subtract 1 from each side
0 = cos^2(x)
Take the square root of each side
0 = cos(x)
x = pi/2, 3pi/2
But since we took the square root, we need to verify the solutions in the original equation
tan(x), sec^2 (x) cannot have cos(x) be zero so these solutions will not work
There are no solutions
No solutions