Siny=sin (a+y)x then proof that dy/dx=sin a /(1-2xcosa+x^2)how?

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Mar 8, 2018

Answer:

We use
#color(red)((1)d/(dx)(tan^(-1)X)=1/(1+X^2)#
#color(red)((2)d/(dx)(u/v)=(v*u^'-u*v^')/v^2)#

Explanation:

Here,
#siny=sin(a+y)*x=x[sinacosy+cosasiny]#
#=>siny=xsinacosy+xcosasiny#
#=>siny-xcosasiny=xsinacosy#
#=>siny(1-xcosa)=xsinacosy#
#=>siny/cosy=(xsina)/(1-xcosa)##=>tany=(xsina)/(1-xcosa)#
#=>y=tan^(-1)((xsina)/(1-xcosa))#
#=>(dy)/(dx)=1/(1+((xsina)/(1-xcosa))^2)d/(dx)((xsina)/(1-xcosa))#
#=(cancel((1-xcosa)^2))/((1-xcosa)^2+(xsina)^2)((1-xcosa)*sina-x*sina*(-cosa))/(cancel((1-xcosa)^2))#
#=(sina-xcosasina+xsinacosa)/(1-2xcosa+x^2cos^2a+x^2sin^2a)=(sina)/(1-2xcosa+x^2(cos^2a+sin^2a))=(sina)/(1-2xcosa+x^2#

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