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Question

If x cos(a+y) = cos y, then prove that dydx=cos2(a+y)sina

Hence show that sin a d2ydx2+sin2(a+y)dydx=0

OR Find dydx if y=sin1[6x414x25].

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Solution

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

Diff. w.r.t. y both sides, we get dxdy=cos(a+y)(sin y)cos y(sin(a+y))cos2(a+y)

dxdy=cos y sin(a+y)cos(a+y)sin ycos2(a+y)dxdy=sin(a+yy)cos2(a+y) dydx=cos2(a+y)sin a.

Now sin adydx=cos2(a+y) sin ad2ydx2=2cos(a+y)sin(a+y)dydx

sin ad2ydx2=sin2a+ydydx sin ad2ydx2+sin2(a+y)dydx=0.

OR

We have y=sin1[6x414x25]. Put 2x=sin θθ=sin12x.....(i)

y=sin1[3 sin θ41sin2 θ5]. y=sin1[35sin θ45cos θ]Let 35=cos αsin α=45...(ii) y=sin1[cos α sin θsin α cos θ]y=sin1[sin(θα)]=θα y=sin12xsin145[By(i)and(ii)]dydx=214x20=214x2


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