Which of the following lie in the span of the set{cos2(x),sin2(x)}? Select all that apply. A. tan2(x) B. cos(2x) C. sin2(2x) D. sin2(x)+1
Question
Which of the following lie in the span of the set {cos2(x), sin2(x)}?
Select all that apply.
A. tan2(x)
B. cos(2x)
C. sin2(2x)
D. sin2(x) + 1
Solution
To determine which of the given options lie in the span of the set {cos2(x), sin2(x)}, we need to check if each option can be expressed as a linear combination of cos2(x) and sin2(x).
A. tan2(x): This option cannot be expressed as a linear combination of cos2(x) and sin2(x). Therefore, it does not lie in the span of the given set.
B. cos(2x): This option can be expressed as a linear combination of cos2(x) and sin2(x). Using the double angle identity, cos(2x) = cos2(x) - sin2(x). Therefore, it lies in the span of the given set.
C. sin2(2x): This option cannot be expressed as a linear combination of cos2(x) and sin2(x). Therefore, it does not lie in the span of the given set.
D. sin2(x) + 1: This option can be expressed as a linear combination of cos2(x) and sin2(x). sin2(x) + 1 = 1 + sin2(x), which is equivalent to cos2(x). Therefore, it lies in the span of the given set.
In summary, the options that lie in the span of the set {cos2(x), sin2(x)} are B. cos(2x) and D. sin2(x) + 1.
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