alleryStart learning →Which expression is not one of the three standard forms of cos(2x)?
Acos^2 x - sin^2 x
B2 cos^2 x - 1
C1 - 2 sin^2 x
D2 sin^2 x - 1
Answer & Solution
Correct answer: D. 2 sin^2 x - 1
1. Setting both angles equal in the cosine sum formula gives cos(2x) = cos^2 x - sin^2 x.
2. Replacing cos^2 x by 1 - sin^2 x turns that into 1 - 2 sin^2 x.
3. Replacing sin^2 x by 1 - cos^2 x turns it into 2 cos^2 x - 1.
4. Those are the three standard forms, so 2 sin^2 x - 1 is the odd one out.
5. In fact 2 sin^2 x - 1 is the negative of cos(2x), so a student who picks it has flipped a sign.
_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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