Question

siny=xsin(a+y) then prove dy/dx= sina/(1-2xcosa+a^2)

Answer 1

Given $\sin y=x\sin (a+y)$. Expand this equation,

$\sin y=x\sin (a)\cos y+x\sin y \cos a\\ \sin y(1-x\cos a)=x \sin a \cos y\\ \tan y=\frac{x \sin a}{1-x\cos a}$

Now differentiate the equation with respect to $x$,

$\frac{d}{dx}\tan y=\frac{d}{dx}\frac{x \sin a}{1-x\cos a} \\ \sec^2 y \frac{dy}{dx}=\frac{(1-x\cos a) \sin a+x \sin a\cos a}{(1-x\cos a)^2} \\ (1+\tan^2 y )\frac{dy}{dx}=\frac{(1-x\cos a) \sin a+x \sin a\cos a}{(1-x\cos a)^2} \\ (1+\frac{x^2 \sin^2 a}{(1-x\cos a)^2} )\frac{dy}{dx}=\frac{(1-x\cos a) \sin a+x \sin a\cos a}{(1-x\cos a)^2} \\ (\frac{1-2x\cos a+x^2 }{(1-x\cos a)^2} )\frac{dy}{dx}=\frac{\sin a}{(1-x\cos a)^2}$\\

$\frac{dy}{dx}=\frac{\sin a}{1-2x\cos a+x^2} =RHS$.

The proof is complete.

Answer 2

Answer:

Given: sin y = x sin (a+y)

To show: dy/dx = sina/(1 - 2x cos a + x²)

First we simplify the given statement,

sin y = x (sin a . cos y + cos a . sin y)

sin y = x sin a . cos y + x cos a. sin y

sin y - x sin y . cos a = x cos y. sin a

sin y (1- x cos a) = x cos y. sin a

sin y = x cos y. sin a / (1- x cos a)

sin y / cos y = x.sin a / (1 - x cos a)

tan y = x.sin a / (1 - x cos a)

Now find derivate of both sides,

$\frac{\mathrm{d}(tan\,y)}{\mathrm{d}x}=\frac{\mathrm{d}(\frac{x\,sin\,a}{1-x\,cos\,a})}{\mathrm{d}x}$

$sec^2\,y\:\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{(1-x\,cos\,a)\frac{\mathrm{d}(x\,sin\,a)}{\mathrm{d}x}-(x\,sin\,a){\frac{\mathrm{d}(1-x\,cos\,a)}{\mathrm{d}x}}}{(1-x\,cos\,a)^2}$

$(1-tan^2\,y)\:\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{(1-x\,cos\,a)\,sin\,a-(x\,sin\,a)(-cos\,a)}{(1-x\,cos\,a)^2}$

$(1-(\frac{x\,sin\,a}{1-x\,cos\,a})^2)\:\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{sin\,a}{(1-x\,cos\,a)^2}$

[tex]\frac{1+x^2\,cos^2\,a-2x\,cos\,a-x^2\,sin^2\,a}{(1-x\,cos\,a)^2}\:\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{sin\,a}{(1-x\,cos\,a)^2} [/tex]

[tex]\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{sin\,a}{(1-x\,cos\,a)^2}\times\frac{(1-x\,cos\,a)^2}{1-2x\,cos\,a+x^2(cos^2\,a+\,sin^2\,a)} [/tex]

Therefore, dy/dx = sina/(1 - 2x cos a + x²)

Hence Proved.