Question

If siny=x(sin(a+y)) then prove that dy/dx=sin^2(a+y)/sina

Answer 1

Answer:

Recall:

sin(A-B) = sin A cos B - cos A sin B       ... (1)

Then...

sin y = x sin (a+y)         ... (2)

=> cos y dy = x cos(a+y) dy + sin(a+y) dx     [ differentiated both sides ]

=> ( cos y - x cos(a+y) ) dy = sin(a+y) dx       [ rearranged ]

=> ( sin(a+y) cos y - x sin(a+y) cos(a+y) ) dy = sin²(a+y) dx  [ multiplied both sides by sin(a+y) ]

=> ( sin(a+y) cos y - sin y cos(a+y) ) dy = sin²(a+y) dx        [ used eqn (2) ]

=> sin( (a+y) - y ) dy = sin²(a+y) dx           [ used identity (1) ]

=> sin a dy = sin²(a+y) dx                       [ simplified ]

=> dy / dx = sin²(a+y) / sin a