alleryStart learning →Write 25 - 9 sin^2 x as a product using the difference of squares.
A(5 - 3 sin x)(5 + 3 sin x)
B(5 - 9 sin x)(5 + 9 sin x)
C(25 - 3 sin x)(1 + 3 sin x)
D(5 - 3 sin x)(5 - 3 sin x)
Answer & Solution
Correct answer: A. (5 - 3 sin x)(5 + 3 sin x)
1. Read the expression as a^2 - b^2 with a^2 = 25 and b^2 = 9 sin^2 x.
2. Then a = 5 because 5^2 = 25.
3. And b = 3 sin x because (3 sin x)^2 = 9 sin^2 x.
4. The difference of squares formula a^2 - b^2 = (a - b)(a + b) now gives (5 - 3 sin x)(5 + 3 sin x).
5. Expanding that product returns 25 - 9 sin^2 x, so the factorisation checks out.
6. Using 9 sin x inside the brackets squares to 81 sin^2 x, which is nine times too large.
7. Repeating the same bracket would give 25 - 30 sin x + 9 sin^2 x, which has a middle term and the wrong sign on the square.
_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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