If x sin a-y-sin a -cos a-y-0- then prove thatd y-d x-sin 2a-y-sin a

We have, x sin(a+y)+sin a.cos(a+y)=0 ⇒ x sin(a+y) = sin a.cos(a+y) ⇒ x= sin a.cos(a+y)/sin(a+y) ⇒ x= sin a.cot(a+y) ∴ dx/dy= sin a.[ cosec^2(a+y)].d/dy(a+y) dx ...

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Standard XII

Mathematics

Sign of Trigonometric Ratios in Different Quadrants

If x sin a+y+...

Question

If $x s i n (a + y) + s i n a . c o s (a + y) = 0$ , then prove that
$\frac{d y}{d x} = \frac{s i n^{2} (a + y)}{s i n a}$

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Solution

We have,
$x s i n (a + y) + s i n a . c o s (a + y) = 0 \Rightarrow x s i n (a + y) = - s i n a . c o s (a + y) \Rightarrow x = \frac{- s i n a . c o s (a + y)}{s i n (a + y)} \Rightarrow x = - s i n a . c o t (a + y) ∴ \frac{d x}{d y} = - s i n a . [- c o s e c^{2} (a + y)] . \frac{d}{d y} (a + y) \frac{d x}{d y} = s i n a . \frac{1}{s i n^{2} (a + y)} ∴ \frac{d y}{d x} = \frac{s i n^{2} (a + y)}{s i n a}$

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If x sin(a+y)+sin a.cos(a+y)=0, then prove that dydx=sin2(a+y)sina

We have, x sin(a+y)+sin a.cos(a+y)=0⇒x sin(a+y)=−sin a.cos(a+y)⇒x=−sin a.cos(a+y)sin(a+y)⇒x=−sin a.cot(a+y)∴ dxdy=−sin a.[−cosec2(a+y)].ddy(a+y)dxdy=sin a.1sin2(a+y)∴ dydx=sin2(a+y)sina

If $x s i n (a + y) + s i n a . c o s (a + y) = 0$ , then prove that
$\frac{d y}{d x} = \frac{s i n^{2} (a + y)}{s i n a}$