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Question

If xcos(a+y)=cos y then prove that

dydx=cos2(a+y)sin a.

Hence, show that sin ad2ydx2+sin 2(a+y)dydx=0.



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Solution

Given:
x cos(a+y)=cos y

x=cos ycos(a+y) .............(1)

Differentiating equation (1) w.r.t.y

dxdy=sin y[cos(a+y)]cos y[sin(a+y)]cos2(a+y)

dxdy=sin y(cos(a+y))+sin(a+y).cos ycos2(a+y)

dxdy=sin(a+y).cos ysin y.cos(a+y)cos2(a+y)

dxdy=sin(a+yy)cos2(a+y)

dxdy=sin acos2(a+y)

dydx=cos2(a+y)sin a .................(2)

Now we have sin adydx=cos2(a+y) ...............(3)

Differentiating equation (3) w.r.t. x, we will get

sin ad2ydx2=2cos(a+y)[sin(a+y)]dydx

sin ad2ydx2=2 sin(a+y)cos(a+y).dydx

sin ad2ydx2=sin 2(a+y)dydx

sin ad2ydx2+sin 2(a+y)dydx=0

Hence Proved.

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