prove this identity

**zodiacbrave** · September 13th 2009, 05:18 PM

2sin(x)/[(cos(x)+sin(x)] = tan(2x) + 1 - sec(2x)

First, I turn tan(2x) into [sin(2x) / cos(2x)] then I turn sec(2x) into 1/cos(2x)

So, I end up with... [sin(2x) / cos(2x)] + 1 - (1/(cos2x))

I multiple and get the GCD of cos(2x)..

[sin(2x) + cos(2x) - 1] / cos(2x)

Now things get kind of fuzzy.. I probably didn't do this the best way thus far but I can't really advance much further.. I need help, please.

**VonNemo19** · September 13th 2009, 05:59 PM

Originally Posted by zodiacbrave

2sin(x)/[(cos(x)+sin(x)] = tan(2x) + 1 - sec(2x)

First, I turn tan(2x) into [sin(2x) / cos(2x)] then I turn sec(2x) into 1/cos(2x)

So, I end up with... [sin(2x) / cos(2x)] + 1 - (1/(cos2x))

I multiple and get the GCD of cos(2x)..

[sin(2x) + cos(2x) - 1] / cos(2x)

Now things get kind of fuzzy.. I probably didn't do this the best way thus far but I can't really advance much further.. I need help, please.

Start with right side with the identy

$tan(2x)=\frac{2tanx}{1-tan^2x}$ and $sec(2x)=\frac{1}{cos(2x)}=\frac{1+tan^2x}{1-tan^2x}$

Then the right side reduces to

$\frac{2tanx+2}{1-tan^2x}$

Chage tan to terms of sin and cos and you should be able to get there.

**Soroban** · September 13th 2009, 07:31 PM

Hello, zodiacbrave!

Another approach . . .

Prove: . $\frac {2\sin x}{\cos x+\sin x} \:=\: \tan(2x) + 1 - \sec(2x)$

Multiply the left side by: . $\frac{\cos x - \sin x}{\cos x - \sin x}$

$\frac{2\sin x}{\cos x + \sin x}\cdot\frac{\cos x - \sin x}{\cos x - \sin x}$

. . $=\;\frac{\overbrace{2\sin x\cos x}^{\sin(2x)} - \overbrace{2\sin^2\!x}^{1-\cos(2x)}}{\underbrace{\cos^2\!x - \sin^2\!x}_{\cos(2x)}}$

. . $= \;\frac{\sin(2x) + \cos(2x) - 1}{\cos(2x)}$

. . $=\;\frac{\sin(2x)}{\cos(2x)} + \frac{\cos x}{\cos x} - \frac{1}{\cos(2x)}$

. . $=\;\tan(2x) + 1 - \sec(2x)$

**pacman** · September 13th 2009, 11:21 PM

Soroban's LATEXing technique is superb and very instructive, thanks MHF expert

Thread: prove this identity

LinkBack

Thread Tools

Display

prove this identity

Similar Math Help Forum Discussions

prove identity help!

prove the identity

Prove this identity?

Prove that each of the following is an identity

Prove the Identity

Search Tags