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If log tanx=y then prove that sinh(n+1)y+sinh(n1)y=2sinh ny.cosec2x
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**To prove if lntanx=y 

sinh(n+1)y+sinh(n1)y=2sinhnycosec2x** 

Now, as lntanx=y we can write,

ey=tanx,

Also by defination of hyperbolic functions

sinhx=exex2

Hence,

L.H.S. = sinh(n+1)y+sinh(n1)y

=e(n+1)ye(n+1)y2+e(n1)ye(n1)y2=enyeyenyey2+enyeyenyey2=eny(ey+ey2)eny(ey+ey2)=(ey+ey)(enyeny2)=(tanx+cotx)sinhny

=(sinxcosx+cosxsinx)sinhny=(sin2x+cos2xsinxcosx)sinhny=(1sin2x2)sinhny=(2sin2x)sinhny=2sinhnycosec2x

=R.H.S

Hence Proved.

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