alleryStart learning →Rewrite 4 cos^2 x - 1 using the difference of squares.
A(4 cos x - 1)(cos x + 1)
B(2 cos x - 1)(2 cos x - 1)
C(4 cos x + 1)(cos x - 1)
D(2 cos x - 1)(2 cos x + 1)
Answer & Solution
Correct answer: D. (2 cos x - 1)(2 cos x + 1)
1. Both the coefficient 4 and the factor cos^2 x are perfect squares, so 4 cos^2 x = (2 cos x)^2.
2. The number 1 is also a perfect square, so the expression reads (2 cos x)^2 minus 1^2.
3. Apply a^2 - b^2 = (a - b)(a + b) with a = 2 cos x and b = 1.
4. That gives (2 cos x - 1)(2 cos x + 1).
5. Squaring the same bracket twice would give 4 cos^2 x - 4 cos x + 1, which carries a middle term the original does not have.
_Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.1 Solving Trigonometric Equations with Identities_
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