Question

# If sinxsiny=12,cosxcosy=32, where x,y∈(0,π2), then the value of tan(x+y) is equal to:

A
13
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B
14
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C
17
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D
15
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Solution

## The correct option is D √15As sinxsiny=12 and cosxcosy=32, then,tanxtany=sinxcosxsinycosy=sinxsiny×cosycosx=12×23=13tany=3tanxtan(x+y)=tanx+tany1−tanxtanytan(x+y)=tanx+3tanx1−tanx×3tanxtan(x+y)=4tanx1−3tan2x (1)Since, sinxsiny=12 and cosxcosy=32, then,siny=2sinx and cosy=23cosxsin2y=4sin2x and cos2y=49cos2xApplying the trigonometric identity,sin2y+cos2y=14sin2x+49cos2x=14cos2x(sin2xcos2x+19)=19tan2x+19=14cos2x9tan2x+19=sec2x49tan2x+19=1+tan2x436tan2x+4=9+9tan2x27tan2x−5=0tan2x=527tanx=√53√3Put the value of tanx in equation (1),tan(x+y)=4(√53√3)1−3(√53√3)2=4√53√31−3(527)=3√5√3=√15

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