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If cos y = x cos (a + y), with cos a!=+-...

If `cos y = x cos (a + y)`, with `cos a!=+-1`, prove that `(dy)/(dx)=(cos^2(a+y)/(sina))`.

Text Solution

Verified by Experts

Given

`cos y = x cos (a + y)`

`cosy=xcos(a+y)`

`cosy/cos(a+y)=x`

`x=cosy/cos(a+y)`

Differentiating both sides w.r.t. x

`(d(x))/dx=d/dx(cosy/cos(a+y))`


`1=d/dx(cosy/cos(a+y)).dy/dx`

Using quotient rule

`1=(((d(cosy))/dy.cos(a+y)-(d(cos(a+y)))/dy.cosy)/(cos^2(a+y))).dy/dx`

`1=((-siny.cos(a+y)-(-sin(a+y))(d(a+y))/dy.cosy)/(cos^2(a+y))).dy/dx`

`1=((-siny.cos(a+y)+sin(a+y)(0+1).cosy)/(cos^2(a+y))).dy/dx`

`1=((sin(a+y).cosy-cos(a+y).siny)/(cos^2(a+y))).dy/dx`

`1=(sin((a+y)-y))/(cos^2(a+y)).dy/dx`

`(cos^2(a+y))/(sin(a))=dy/dx`

`dy/dx=(cos^2(a+y))/(sin(a))`
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