Final answer:
The equation cos²x = √2/2 has solutions at x = π/4 and x = 3π/4. The general solution is x = π/4 + nπ/2, where n is any integer, which matches option (b).
Step-by-step explanation:
To find all solutions to the equation cos²x = √2/2, we first take the square root of both sides to obtain cos x = ±√(√2/2), which simplifies to cos x = ± 1/√2. This is equivalent to cos x = ± √2/2 since 1/√2 equals √2/2. The values of x where the cosine function equals ±√2/2 are at x = π/4 and x = 3π/4 within the interval [0, 2π]. To find all solutions, we can add multiples of π/2 to these base angles since the period of the cosine function is 2π.
Therefore, the general solution for when cos²x = √2/2 is x = π/4 + nπ/2 and x = 3π/4 + nπ/2, where n represents an integer. This corresponds to option (b) x = π/4 + nπ/2.