What are the solutions of the equation cos^2(x) - 1/2 = 0?

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Question: What are the solutions of the equation cos^2(x) - 1/2 = 0?

Options:

Correct Answer: x = π/4

Solution:

The solutions are x = π/4, 3π/4, 5π/4, and 7π/4.

What are the solutions of the equation cos^2(x) - 1/2 = 0?

What are the solutions of the equation cos^2(x) - 1/2 = 0?

Step 1: Start with the equation cos^2(x) - 1/2 = 0.
Step 2: Add 1/2 to both sides of the equation to isolate cos^2(x).
Step 3: The equation now looks like cos^2(x) = 1/2.
Step 4: Take the square root of both sides to find cos(x). Remember to consider both the positive and negative roots.
Step 5: This gives us cos(x) = √(1/2) or cos(x) = -√(1/2).
Step 6: Simplify √(1/2) to 1/√2, which is the same as √2/2.
Step 7: Now we have two equations: cos(x) = √2/2 and cos(x) = -√2/2.
Step 8: Find the angles x that satisfy cos(x) = √2/2. These angles are x = π/4 and x = 7π/4.
Step 9: Find the angles x that satisfy cos(x) = -√2/2. These angles are x = 3π/4 and x = 5π/4.
Step 10: Combine all the solutions: x = π/4, 3π/4, 5π/4, and 7π/4.

Trigonometric Equations – The question tests the ability to solve a trigonometric equation involving the cosine function.
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