Final answer:
To solve sin^2(θ) - cos^2(θ) = 0, use the trigonometric identity sin^2(θ) + cos^2(θ) = 1 and rearrange the equation. Divide by -2, take the square root, and identify the possible solutions using the unit circle.
Step-by-step explanation:
To solve the equation sin^2(θ) - cos^2(θ) = 0, we can use the trigonometric identity sin^2(θ) + cos^2(θ) = 1. By rearranging the equation, we get sin^2(θ) = 1 - cos^2(θ). Substituting this into the original equation, we have 1 - cos^2(θ) - cos^2(θ) = 0.
Combine like terms: -2cos^2(θ) = -1
Divide both sides by -2: cos^2(θ) = 1/2
Take the square root of both sides: cos(θ) = ±√(1/2)
From the unit circle, we know that cos(θ) = √(2)/2 and cos(θ) = -√(2)/2 for angles of θ = π/4 + 2nπ and θ = 3π/4 + 2nπ respectively, where n is an integer. To solve the equation sin2θ - cos2θ = 0, we can use a trigonometric identity. The key identity we'll utilize is the Pythagorean identity, which states that sin2θ + cos2θ = 1. This can be rewritten as sin2θ = 1 - cos2θ, which allows us to substitute in the original equation.
Substituting 1 - cos2θ for sin2θ in the original equation gives us:
1 - cos2θ - cos2θ = 0
This simplifies to:
1 - 2cos2θ = 0
From here, we can solve for cos2θ:
2cos2θ = 1
cos2θ = 1/2
Now we take the square root of both sides, giving us two possible solutions:
cos θ = ±√(1/2)
This corresponds to angles for which the cosine value is ±√(1/2). Depending on the range of θ, these angles can be found in the first and second quadrants (for +√(1/2)) and first and fourth quadrants (for -√(1/2)), typically at 45° and 135°, or π/4 and 3π/4 radians, respectively.