# Trig identities

• Feb 20th 2008, 02:04 AM
gracey
Trig identities
1 / cos^4ф - 1 / cos^2ф = tan^4ф + tan^2ф
• Feb 20th 2008, 02:25 AM
mr fantastic
Quote:

Originally Posted by gracey
cos^4 ф - sin^4 ф + 1 = 2cos^2ф

Mr F says: Note that $\displaystyle \cos^4 \phi - \sin^4 \phi = (\cos^2 \phi + \sin^2 \phi)(\cos^2 \phi - \sin^2 \phi) = \cos^2 \phi - \sin^2 \phi$
[snip]

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• Feb 20th 2008, 02:36 AM
mr fantastic
Quote:

Originally Posted by gracey
[snip]
1 / cos^4ф - 1 / cos^2ф = tan^4ф + tan^2ф

Right Hand Side $\displaystyle \, =\tan^4 \phi + \tan^2 \phi = \frac{\sin^4 \phi}{\cos^4 \phi} + \frac{\sin^2 \phi}{\cos^2 \phi}$.

Now note that:

$\displaystyle \frac{\sin^4 \phi}{\cos^4 \phi} = \frac{(\sin^2 \phi)^2}{\cos^4 \phi} = \frac{(1 - \cos^2 \phi)^2}{\cos^4 \phi} = \frac{1 - 2 \cos^2 \phi + \cos^4 \phi}{\cos^4 \phi} = \frac{1}{\cos^4 \phi} - \frac{2}{\cos^2 \phi} + 1$.

$\displaystyle \frac{\sin^2 \phi}{\cos^2 \phi} = \frac{1 - \cos^2 \phi}{\cos^2 \phi} = \frac{1}{\cos^2 \phi} - 1$.

Therefore the right hand side becomes .......
• Mar 6th 2008, 11:36 PM
mrbuttersworth
I did this one differently. I went like...

left side
step one multiply$\displaystyle \frac{1}{cos^2x}$ by $\displaystyle \frac{cos^2x}{cos^2x}$ and then add because we've achieved a common denominator.
Then using the pythagorean identity, we can change $\displaystyle 1-cos^2x$into $\displaystyle sin^2x$

Right side
convert the tan's into their proper sin and cosine values.
multiply $\displaystyle {sin^2x}{cos^2x}$ by $\displaystyle \frac{cos^2x}{cos^2x}$ and add because we've achieved a common denominator.
factor out a $\displaystyle sin^2x$
use the pythagorean identity to turn $\displaystyle sin^2x+cos^2x$ into 1
finally we end up with $\displaystyle \frac{sin^2x}{cos^4x}$

L=R
• Mar 6th 2008, 11:37 PM
mrbuttersworth
Also, I would like to know how to write equations neater and more aesthetically pleasing than the method i'm using now. How do you guys do it?
• Mar 6th 2008, 11:59 PM
mr fantastic
Quote:

Originally Posted by mrbuttersworth
Also, I would like to know how to write equations neater and more aesthetically pleasing than the method i'm using now. How do you guys do it?