Trigonometry Derivative ('Show That' Question)

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May 8th 2008, 06:38 AM
Simplicity

Trigonometry Derivative ('Show That' Question)

Q: Given that $y= \mathrm{arctan} \frac{x}{a}$ , where $a$ is a constant, show that $x \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{\tan y}{1 + \tan ^2 y}$ .

So... I have done (and reached a dead end):

$y= \mathrm{arctan} \frac{x}{a}$
$\therefore \tan y = \frac {x}{a}$
$\frac{\tan y}{1 + \tan ^2 y}$
$= \frac{(\frac{x}{a})}{1 + (\frac{x}{a})^2}$
$= \frac{(\frac{x}{a})}{\frac{a^2 + x^2}{a^2}}$
$=\frac{ax}{a^2 + x^2}$

But That doesn't lead me anywhere. Please help. Thanks in advance.
May 8th 2008, 06:48 AM
Gusbob

$y = arctan(\frac{x}{a})$

From the table of standard integrals, we know that

$\int \frac{1}{a^2+b^2}\,dx = \frac{1}{a}arctan(\frac{x}{a})$
But $\frac{1}{a}= 1$ in our equation. Therefore a = 1 (not too sure about this step. I think if you went forward and worked with x/a it will cancel out anyway)

$y = arctan(\frac{x}{a})$
$\frac{dy}{dx} = \frac{1}{1^2+ x^2}$
Multiplying both sides by x
$x\frac{dy}{dx} = \frac{x}{1+x^2}$
But $x = tan (y)$
$x\frac{dy}{dx} = \frac{tan (y)}{1+ tan^2(y)}$

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