alleryStart learning →Expanding (sin x + cos x)^2 and applying a Pythagorean identity yields which expression?
A1 + sin(2x)
B1 - sin(2x)
C1 + cos(2x)
D2 + sin(2x)
Answer & Solution
Correct answer: A. 1 + sin(2x)
1. The perfect square formula gives (sin x + cos x)^2 = sin^2 x + 2 sin x cos x + cos^2 x.
2. Group the two squares together as (sin^2 x + cos^2 x) + 2 sin x cos x.
3. The bracket equals 1 by the Pythagorean identity.
4. The remaining term 2 sin x cos x is exactly sin(2x).
5. So the expression equals 1 + sin(2x), and the option 1 - sin(2x) would come from squaring the difference instead of the sum.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.3 Double-Angle, Half-Angle, and Reduction Formulas_
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