Solve the equation for 0 x 2pi tan2x secx 1Solutiontan2x

Solve the equation for 0 < x < 2pi

tan²x - secx = 1

we have sec²x - tan²x = 1 ==> tan²x = sec²x -1

==> tan²x - secx = 1 ==> (sec²x -1) - secx = 1

==> sec²x - secx - 2 = 0

==> sec²x + secx - 2secx - 2 = 0

==> secx(secx + 1) -2(secx + 1) = 0

==> (secx + 1)(secx -2) = 0

==> secx = -1 , secx = 2

==> cosx = -1 , cosx = 1/2

==> x = , x = /3 , 5/3 (since cosx is positive in I and IV Quadrant)

Hence x = /3 , , 5/3