Final answer:
To solve for θ in the equation sec(-2θ) = csc(-2θ), we can convert the functions to cosine and sine, and use the property of inverse trigonometric functions. The only solution in the given range is θ = 0.
Step-by-step explanation:
To solve the equation sec(-2θ) = csc(-2θ), we can use the reciprocal identities of trigonometric functions. First, we need to convert sec(-2θ) and csc(-2θ) to their equivalent cosine and sine functions. The reciprocal identity of secant is cosine, and the reciprocal identity of the cosecant is sine. So, we have cos(-2θ) = sin(-2θ).
Next, since the angles are equal, we can use the property of inverse trigonometric functions which states that if sin(a) = sin(b), then a = b + 2πn or a = π - b + 2πn, where n is an integer. In this case, we have -2θ = -2θ + 2πn.
Simplifying the equation, we get 0 = 2πn. Since 0 is less than π, the only solution in the given range of 0 ≤ θ < π is θ = 0.