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Question

Determine the smallest positive value of x (in degrees) for which tan(x+100o)=tan(x+50o)(tanx)tan(x−50o)

A
30o
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B
45o
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C
60o
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D
75o
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Solution

The correct option is A 30o
Given tan(x+100o)=tan(x+50o)tanxtan(x50o)
tan(x+100o)tan(x50o)=tan(x+50o)tanx
sin(x+100o)cos(x500)cos(x+100o)sin(x500)=sin(x+50o)sinxcos(x+50o)cosx
Applying componendo and dividendo , we get
sin(2x+50o)sin1500=cos50ocos(2x+50o)
sin(2x+50o)cos(2x+50o)=sin1500cos50o
sin(4x+100o)=sin400
sin(4x+100o)=sin(400)
4x+100o=nπ(1)n400
Substitute n=1, smallest positive value of x is given by
4x=1200
x=300

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