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If cos x + sin x = a with -pi/2 < x < -pi/4, then cos 2x equals:

1. a*sqrt(2 - a²)
2. a²
3. a*sqrt(2 - a)
4. a*sqrt(2 + a)

Correct answer: a*sqrt(2 - a²)

Solution

Since (cos x + sin x)² = 1 + sin 2x = a², we get (cos x - sin x)² = 2 - a². In the interval (-pi/2, -pi/4), cos x - sin x > 0, so cos 2x = a*sqrt(2 - a²).

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