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Question

If x=sin70o.sin50o and y=cos60o.cos80o, then what is xy equal to?

A
116
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B
18
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C
14
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D
12
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Solution

The correct option is A 116
x=sin(700).sin(500)
We know that sin(θ)=cos(900θ)
x=sin(700).sin(500)=cos(90700).cos(90500)=cos(200).cos(400)

y=cos(600).cos(800)
We know that cos(θ)=sin(900θ)
y=cos(600).cos(800)=sin(90600).sin(90800)=sin(300).sin(100)

xy=cos(200).cos(400).sin(300).sin(100)

Multiply both sides of equation with 2cos(100)

xy(2cos(100))=cos(200).cos(400).sin(300).sin(100).2cos(100)

Now, we will use the formula 2sinθcosθ=sin2θ

xy(2cos(100))=cos(200).cos(400).sin(300).(2sin(100).cos(100))

2xy.cos(100)=cos(200).cos(400).sin(300).(sin200)

Multiply 2 on both sides of above equation

4xy.cos(100)=(2(sin200).cos(200)).cos(400).sin(300)

4xy.cos(100)=(sin(400)).cos(400).sin(300)

Multiply 2 on both sides of above equation

8xy.cos(100)=(2sin(400).cos(400)).sin(300)

8xy.cos(100)=sin(800).sin(300)

8xy.cos(100)=cos(100).sin(300)

8xy=sin(300)=12

xy=116

So, the answer is option (A)

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