Question

Find all solutions in the interval [0, 2π).

sec2x - 2 = tan2x

Answer 1

do you mean Find all solutions sec^2x + tan^2x = 3 in interval [0,2pi)?

Answer 2

yes. Sorry about that.

Answer 3

sec²(x) + tan²(x) = 3

Pythagorean trig identity:
sec²(x) + (sec²(x) - 1) = 3

Combine like terms on left:
2sec²(x) - 1 = 3

Add 1 to both sides:
2sec²(x) = 4

Divide both sides by 2:
sec²(x) = 2

Take the square root of both sides:
sec(x) = ±√2

Put in terms of cosine:
1/cos(x) = ±√2

Multiply both sides by ±cos(x)/√2:
cos(x) = ±1/√2

Multiply right-hand side by √2/√2:
cos(x) = ±√2/2

Use the unit circle to evaluate:
x = π/4, 3π/4, 5π/4, 7π/4