Solve finding all solutions in 0 2 3sec x 2tan2x can you sh

Solve, finding all solutions in [0, 2). 3sec x = 2tan2x can you show work

Solution

we are given 3secx=2tan^2x   so we will apply this indentity we will get

Applying the identity, sec²x - tan²x = 1,
the given equation reduces to: 2sec²x - 3sec(x) - 2 = 0

ii) On factorization, it is: {sec(x) - 2}{2sec(x) + 1} = 0

==> Either sec(x) = 2 or sec(x) = -1/2

As secant function for all real x, has no value in (-1, 1), sec(x) = -1/2 is not possible. Thus only one value is possible, that is sec(x) = 2

==> x = 2n ± (/3), where n is an integer.