If sin(x) = 1/√2, what is tan(x)?

If sin(x) = 1/√2, what is tan(x)?

Step 1: Start with the given equation: sin(x) = 1/√2.
Step 2: Recall the relationship between sine, cosine, and tangent: tan(x) = sin(x) / cos(x).
Step 3: We need to find cos(x). Since sin(x) = 1/√2, we can use the Pythagorean identity: sin²(x) + cos²(x) = 1.
Step 4: Substitute sin(x) into the identity: (1/√2)² + cos²(x) = 1.
Step 5: Calculate (1/√2)², which is 1/2. So, we have 1/2 + cos²(x) = 1.
Step 6: Solve for cos²(x): cos²(x) = 1 - 1/2 = 1/2.
Step 7: Take the square root of both sides to find cos(x): cos(x) = 1/√2 (we take the positive root assuming x is in the first quadrant).
Step 8: Now substitute sin(x) and cos(x) back into the tangent formula: tan(x) = (1/√2) / (1/√2).
Step 9: Simplify the fraction: (1/√2) / (1/√2) = 1.
Step 10: Therefore, tan(x) = 1.

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