# Why does tan^2 y = x

• Aug 28th 2012, 05:06 PM
Furyan
Why does tan^2 y = x^2
Hello,

I continue to struggle with trig.

The question is:

Given that $x = \tan y$ show that $\dfrac{dy}{dx} = \dfrac{1}{1 + x^2}$.

The answer given, most of which I follow, is:

$\dfrac{d}{dy} (\tan y) = \sec^2 y$

Taking the reciprocal of both sides.

$\dfrac{dy}{dx} = \dfrac{1}{sec^2 y}$

$\dfrac{dy}{dx} = \dfrac{1}{1 + \tan^2 y}$

Hence, and this is the bit I don't get.

$\dfrac{dy}{dx} = \dfrac{1}{1 + x^2}$

It's bound to be something I'm supposed to know, something basic, but why is $\tan^2 y = x^2$

Thanks
• Aug 28th 2012, 05:34 PM
topsquark
Re: Why does tan^2 y = x
Quote:

Originally Posted by Furyan
Hello,

I continue to struggle with trig.

The question is:

Given that $x = \tan y$ show that $\dfrac{dy}{dx} = \dfrac{1}{1 + x^2}$.

The answer given, most of which I follow, is:

$\dfrac{d}{dy} (\tan y) = \sec^2 y$

Taking the reciprocal of both sides.

$\dfrac{dy}{dx} = \dfrac{1}{sec^2 y}$

$\dfrac{dy}{dx} = \dfrac{1}{1 + \tan^2 y}$

You have been given $x = tan(y)$. Then by squaring both sides: $tan^2(y) = x^2$.

-Dan