If y=cos2(45∘+x)+(sinx−cosx)2, where x∈(0,π2], then the possible value(s) of y is/are
A
0
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B
32
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C
3
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D
1
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Solution
The correct option is D1 cos2(45∘+x)+(sinx−cosx)2=(cos45∘cosx−sin45∘sinx)2+(1−sin2x) =[cosx√2−sinx√2]2+(1−sin2x)=(1−sin2x)2+(1−sin2x) =32(1−sin2x)
As, 0<x≤π2⇒0<2x≤π⇒0≤sin2x≤1⇒0≥−sin2x≥−1⇒−1≤−sin2x≤0⇒0≤1−sin2x≤1⇒0≤32(1−sin2x)≤32∴y∈[0,32]