Explanation
To find the range of A, we simplify the expression:
1. Simplify the Expression
Substitute cos2θ=1−sin2θ:
2. Perfect Square Method Let t=sin2θ. Since 0≤sin2θ≤1, we have 0≤t≤1. The expression becomes:
To find the minimum and maximum, we complete the square:
3. Finding the Maximum and Minimum Values
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Minimum Value: Occurs when (t−21)2=0, i.e., t=21.
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Maximum Value: Occurs at the boundaries of t∈[0,1]. If t=0: A=(0−21)2+43=41+43=1. If t=1: A=(1−21)2+43=41+43=1.
Conclusion: The range of A is 43≤A≤1.
Correct Option: (d) 43≤A≤1
