Final answer:
To solve the equation sec²x + 2tanx = 0, apply the Pythagorean trigonometric identity and simplify to find the solutions for x.
Step-by-step explanation:
To solve the trigonometric equation sec²x + 2tanx = 0, we can use trigonometric identities to simplify the equation. Since sec²x is equal to 1 + tan²x (the Pythagorean identity), we can substitute in this identity and set the equation to zero to find the values of x that satisfy the equation. This will require factoring and using trigonometric properties to find the solutions within the desired interval.