Question

How do you solve `tan^2x+4=2sec^2x+tanx` in the interval [0,360]?

Answer 1

Use the identity `sec^2theta = 1 + tan^2theta`.

`tan^2x + 4 = 2(1 + tan^2x) + tanx`

`tan^2x + 4 = 2 + 2tan^2x + tanx`

`0 = tan^2x + tanx - 2`

`0 = tan^2x + 2tanx - tanx - 2`

`0 = tanx(tanx + 2) - 1(tanx + 2)`

`0 = (tanx - 1)(tanx + 2)`

`tanx = -2` and `tanx = 1`

`x = 180˚ - arctan(2)`, `360˚- arctan(2), 45˚, 225˚`

Hopefully this helps!