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

Question

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

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Answer

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

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