# A Couple Trig Identities

• Dec 15th 2010, 04:44 PM
~berserk
A Couple Trig Identities
This is the beginning of a couple of identities that i will probably need assistance on, all help given is greatly appreciated.

$\displaystyle \displaystyle{\frac{\tan x - \cot x}{\tan x + \cot x}=2\sin^2x-1}$
first i switched them all to sin and cos
then i found like denominators and combined
then i multiplied by the reciprocal leading me to here:

$\displaystyle \displaystyle{\frac{\sin x - \cos x}{\cos x \sin x} \bullet \frac{\cos x \sin x}{\sin x + \cos x}}$

then got $\displaystyle \displaystyle{\frac{\sin x - \cos x}{\sin x + \cos x} = 2\sin^2x-1}$
I'm unsure where to go from here
• Dec 15th 2010, 04:56 PM
Quote:

Originally Posted by ~berserk
This is the beginning of a couple of identities that i will probably need assistance on, all help given is greatly appreciated.

$\displaystyle \displaystyle{\frac{\tan x - \cot x}{\tan x + \cot x}=2\sin^2x-1}$
first i switched them all to sin and cos
then i found like denominators and combined
then i multiplied by the reciprocal leading me to here:

$\displaystyle \displaystyle{\frac{\sin x - \cos x}{\cos x \sin x} \bullet \frac{\cos x \sin x}{\sin x + \cos x}}$

then got $\displaystyle \displaystyle{\frac{\sin x - \cos x}{\sin x + \cos x} = 2\sin^2x-1}$
I'm unsure where to go from here

If you did the 1st one that way, your squares have gone missing...

$\displaystyle \displaystyle\frac{\left[\frac{sinx}{cosx}-\frac{cosx}{sinx}\right]}{\left[\frac{sinx}{cosx}+\frac{cosx}{sinx}\right]}=\frac{\left[\frac{1}{sinxcosx}\right]}{\left[\frac{1}{sinxcosx}\right]}\;\frac{sin^2x-cos^2x}{sin^2x+cos^2x}$

$\displaystyle =sin^2x-cos^2x$

since $\displaystyle sin^2x+cos^2x=1$

and now use $\displaystyle cos^2x=1-sin^2x$ to finish
• Dec 15th 2010, 04:56 PM
~berserk
also there is this one where I am unsure where to start: $\displaystyle (\sin x + \cos x)^2 = 1 +\sin 2x$
• Dec 15th 2010, 05:01 PM
TheCoffeeMachine
Quote:

Originally Posted by ~berserk
also there is this one where I am unsure where to start: $\displaystyle (\sin x + \cos x)^2 = 1 +\sin 2x$

$\displaystyle (\sin x + \cos x)^2 = \left(\sin^2{x}+\cos^2{x}\right)+2\sin{x}\cos{x} = ...$
• Dec 15th 2010, 05:03 PM
Quote:

Originally Posted by ~berserk
also there is this one where I am unsure where to start: $\displaystyle (\sin x + \cos x)^2 = 1 +\sin 2x$

That's another $\displaystyle sin^2x+cos^2x=1$ identity in disguise.

That's a crucial one to remember!

$\displaystyle (sinx+cosx)(sinx+cosx)=sinx(sinx+cosx)+cosx(sinx+c osx)$

$\displaystyle =sin^2x+sinxcosx+cosxsinx+cos^2x=sin^2x+cos^2x+2si nxcosx$

Finish up with the identity $\displaystyle sin2x=2sinxcosx$