Short Answer 
The process begins by understanding the tangent function and Pythagorean identities, which aid in transforming the equation 2sec¬≤x Рtan‚A¥x = -1. By rewriting it with Pythagorean identities, the equation simplifies to y¬≤ Р2y Р3 = 0, leading to valid solutions for y and ultimately yielding x = ¬±œA/3 + nœA as the final solutions in radians.
Step 1: Understand Tangent Function and Pythagorean Identities
The range of the tangent function includes all real numbers, which is crucial for solving trigonometric equations. Begin by noting the essential Pythagorean identities, which are:
- sin²(θ) + cos²(θ) = 1
- 1 + tan²(θ) = sec²(θ)
- 1 + cot²(θ) = csc²(θ)
Using these identities helps simplify your solutions for trigonometric equations. The given equation is 2sec¬≤x Рtan‚A¥x = -1.
Step 2: Transform the Equation Using Pythagorean Identities
Apply the second Pythagorean identity to rewrite the equation. Substitute sec²x in terms of tan²x:
- Rewrite as: 2(1 + tan¬≤x) Р(tan¬≤x)¬≤ = -1
- Which simplifies to: (tan¬≤x)¬≤ Р2tan¬≤x Р3 = 0
Let y = tan¬≤x and change the equation to: y¬≤ Р2y Р3 = 0. Factor it to find the possible values of y.
Step 3: Solve for x Using Found Values of y
From the factored equation, you derive (y + 1)(y – 3) = 0, leading to y = -1 (not valid) or y = 3 (valid). Hence:
- Since y = tan¬≤x = 3, then tan(x) = ¬±‚ao3.
- Thus, x = tan‚Aª¬π(¬±‚ao3) gives solutions in degrees: x = ¬±60¬∞ + nœA.
Finally, convert degrees to radians to conclude that the solutions are x = ¬±œA/3 + nœA, for n in ‚N§.