How do you find the exact value tan(x-y)tan(x−y) if sinx=8/17,cosy=3/5sinx=817,cosy=35?

1 Answer
Narad T.
Dec 4, 2016

The answer is =-36/77=−3677

Explanation:

We use

cos^2x+sin^2x=1cos2x+sin2x=1

cos^2y+sin^2y=1cos2y+sin2y=1

sin(x-y)=sinxcosy-sinycosxsin(x−y)=sinxcosy−sinycosx

cos(x-y)=cosxcosy+sinxsinycos(x−y)=cosxcosy+sinxsiny

sinx=8/17sinx=817

cosx=sqrt(1-sin^2x)=sqrt(1-64/289)=sqrt(225/289)=15/17cosx=√1−sin2x=√1−64289=√225289=1517

cosy=3/5cosy=35

siny=sqrt(1-cos^2y)=sqrt(1-9/25)=sqrt(16/25)=4/5siny=√1−cos2y=√1−925=√1625=45

tan(x-y)=sin(x-y)/cos(x-y)tan(x−y)=sin(x−y)cos(x−y)

=(sinxcosy-sinycosx)/(cosxcosy+sinxsiny)=sinxcosy−sinycosxcosxcosy+sinxsiny

=(8/17*3/5-4/5*15/17)/(15/17*3/5+8/17*4/5)=817⋅35−45⋅15171517⋅35+817⋅45

=(24/85-60/85)/(45/85+32/85)=2485−60854585+3285

=-36/77=−3677

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