If sin y = x sin (a + y), prove that dy dx = sin^ 2 (a+y) sin a . - Vidyadip

Question

If sin y = x sin (a + y), prove that $\frac{\text{dy}}{\text{dx}}=\frac{\text{sin}^{2}\text{(a+y)}}{\text{sin a}}$.

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Answer

sin y = x sin (a + y) $\Rightarrow$ cos y $\frac{\text{dy}}{\text{dx}}$ = x cos (a + y) $\frac{\text{dy}}{\text{dx}}$ + sin (a + y)
$\therefore\frac{\text{dy}}{\text{dx}}= \frac{\text{sin (a + y)}}{\text{cos y - x cos (a + y)}}$
$\text{x} = \frac{\text{sin y}}{\text{sin (a+y)}}\Rightarrow \frac{\text{dy}}{\text{dx}}=\frac{\text{sin (a+y)}}{\text{cos y -}\frac{\text{sin y}}{\text{sin (a+y)}}.\text{cos (a+y)}}$
$\therefore \frac{\text{dy}}{\text{dx}}= \frac{\text{sin}^{2}\text{(a+y)}}{\text{sin (a+y) cos y - cos (a + y) sin y}} = \frac{\text{sin}^{2}\text{(a + y)}}{\text{sin a}}$.

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