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Which of the following trigonometric expressions simplifies to 1 for all valid values of alpha?

1. (1 - 2*sin²(alpha)) / (2*cot(pi/4 + alpha)*cos²(pi/4 - alpha))
2. sin(pi - alpha) / (sin(alpha) - cos(alpha)*tan(pi/2)) + cos(pi - alpha)
3. 1 / (4*sin²(alpha)*cos²(alpha)) + (1 - tan²(alpha))² / (4*tan²(alpha))
4. (1 + sin(2*alpha)) / (sin(alpha) + cos(alpha))²

Correct answer: (1 + sin(2*alpha)) / (sin(alpha) + cos(alpha))²

Solution

Option 4 simplifies identically to 1 because (sin(alpha) + cos(alpha))² = sin²(alpha) + 2*sin(alpha)*cos(alpha) + cos²(alpha) = 1 + sin(2*alpha), which exactly equals the numerator.

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