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Question

If sin(x+y)sin(xy)=a+bab,, then show that tanxtany=ab,.

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Solution

Given:- sin(x+y)sin(xy)=a+bab
To Prove:- tanxtany=ab
Proof:-
Using trigonomatric identity:
sin(A+B)=sinAcosB+cosAsinB
sin(AB)=sinAcosBcosAsinB
Therefore,
sin(x+y)sin(xy)=a+bab
sinxcosy+cosxsinysinxcosycosxsiny=a+bab
Comparing L.H.S. and R.H.S., we have
a=sinxcosy
b=cosxsiny
ab=sinxcosycosxsiny=tanxcoty=tanxtany
Hence proved.

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