# If xsiny=sin(p+y),p∈R,show that sinpdy/dx+sin^2y=0?

Kindly refer to a Proof in the Explanation.

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Given (x \sin(y) = \sin(p+y)), where (p) is a constant, differentiate both sides with respect to (x), using the product rule on the left side and the chain rule on the right side:

[ \frac{d}{dx} (x \sin(y)) = \frac{d}{dx} (\sin(p+y)) ]

[ \sin(y) + x \cos(y) \frac{dy}{dx} = \cos(p+y) \frac{dy}{dx} ]

[ x \cos(y) \frac{dy}{dx} - \cos(p+y) \frac{dy}{dx} = - \sin(y) ]

[ \frac{dy}{dx} (x \cos(y) - \cos(p+y)) = - \sin(y) ]

[ \frac{dy}{dx} = \frac{- \sin(y)}{x \cos(y) - \cos(p+y)} ]

Now, we need to express (\sin(p)) in terms of (y) and (x). From the given equation (x \sin(y) = \sin(p+y)), rearrange to get:

[ \sin(p) = x \sin(y) - \sin(y) ]

[ \sin(p) = \sin(y) (x - 1) ]

So, differentiate (\sin(p)) with respect to (x):

[ \frac{d \sin(p)}{dx} = \sin(y) \frac{d}{dx} (x - 1) ]

[ \frac{d \sin(p)}{dx} = \sin(y) ]

Substitute this into the expression for (\frac{dy}{dx}):

[ \frac{dy}{dx} = \frac{- \sin(y)}{x \cos(y) - \cos(p+y)} = \frac{- \sin(y)}{x \cos(y) - \cos(p)} ]

Now, multiply both numerator and denominator by (\sin(y)):

[ \frac{dy}{dx} = \frac{- \sin^2(y)}{x \cos(y) \sin(y) - \sin(y) \cos(p)} ]

[ \frac{dy}{dx} = \frac{- \sin^2(y)}{x \sin(y) \cos(y) - \sin(y) \sin(p)} ]

[ \frac{dy}{dx} = \frac{- \sin^2(y)}{\sin(y)(x \cos(y) - \sin(p))} ]

[ \frac{dy}{dx} = \frac{- \sin(y) \sin(y)}{\sin(y)(x \cos(y) - \sin(p))} ]

[ \frac{dy}{dx} = \frac{- \sin(y)}{x \cos(y) - \sin(p)} ]

So, we've shown that ( \frac{dy}{dx} = \frac{- \sin(y)}{x \cos(y) - \sin(p)} ).

Therefore, ( \sin(p) \frac{dy}{dx} + \sin^2(y) = 0 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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