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If (cosx)^y=(tany)^x , prove that (dy)/(...

If `(cosx)^y=(tany)^x` , prove that `(dy)/(dx)=(logtany+ytanx)/(logcosx-xsecy cos e c\ y)`

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Verified by Experts

Given that,
`(cosx)^y=(tany)^x`
`=>log(cosx)^y=log(tany)^x`
`=>y log(cosx)=xlog(tany)`
`=>d/(dx)(y log(cosx))=d/(dx)(xlog(tany))`
`=>(dy)/(dx)*log(cosx)+y*d/(dx)log(cosx)=1* log(tany)+x*d/(dx)log(tany)`
`=>(dy)/(dx)*log(cosx)+y*1/cosx*d/(dx)(cosx)=log(tany)+x*1/tany*d/(dx)(sec^2y)`
`=>(dy)/(dx)log(cosx)+y*(-sinx)/cosx=log(tany)+x *sec^2y/tany * (dy)/(dx)`
`=>(dy)/(dx)log(cosx)-tanx*y=log(tany)+x*1/cos^2y *cosy/siny *(dy)/(dx)`
`=>(dy)/(dx)(log(cosx)-x/(siny*cosy))=log(tany)+y*tanx`
`=>(dy)/(dx)=(log (tany)+y*tanx)/(log(cosx)-x*cosecy *secy)`
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