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Question

If sinθ+2cosθ=1

Whether the given equation 2sinθcosθ=2 is ?

A
True
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B
False
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Solution

The correct option is B False
sinθ+2cosθ=1
Squaring both sides,
sin2θ+4cos2θ+4sinθcosθ=1
1cos2θ+4cos2θ+4sinθcosθ=1
3cos2θ+4sinθcosθ=0
3cos2θ=4sinθcosθ
3cosθ=4sinθ or cosθ=0
tanθ=34,tanθ is in second or fourth quadrant.
since 42+32=52
Hypotenuse=5units
Case:1:
sinθ=45,cosθ=35 when tanθ in second quadrant.
2sinθcosθ=24535=8+35=115
Case2
sinθ=45,cosθ=35 when tanθ in fourth quadrant.
2sinθcosθ=24535=8+35=515=1
Case:3:
cosθ=0 from above
θ=(2n+1)π2
sinθ=nπ
Hence in any of three cases we observe that 2sinθcosθ=2 is false.

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