# Thread: Trig identities

1. ## Trig identities

1 / cos^4ф - 1 / cos^2ф = tan^4ф + tan^2ф

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

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

3. Originally Posted by gracey
[snip]
1 / cos^4ф - 1 / cos^2ф = tan^4ф + tan^2ф
Right Hand Side $\, =\tan^4 \phi + \tan^2 \phi = \frac{\sin^4 \phi}{\cos^4 \phi} + \frac{\sin^2 \phi}{\cos^2 \phi}$.

Now note that:

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

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

Therefore the right hand side becomes .......

4. I did this one differently. I went like...

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

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

L=R

5. 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?

6. 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?
Read this.