Homework help

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June 1st 2008, 07:14 PM
unmcintosh

Homework help

Can somebody help me simplify this?
(cos2x+sin^2x)/(cos^2x)
June 1st 2008, 07:25 PM
topsquark

Quote:

Originally Posted by unmcintosh

Can somebody help me simplify this?
(cos2x+sin^2x)/(cos^2x)

$\frac{cos(2x) + sin^2(x)}{cos^2(x)}$

$= \frac{(2~cos^2(x) - 1) + sin^2(x)}{cos^2(x)}$

Upon some rearranging in the numerator:
$\frac{(cos^2(x) + sin^2(x)) + cos^2(x) - 1}{cos^2(x)}$

$\frac{1 + cos^2(x) - 1}{cos^2(x)}$

$\frac{cos^2(x)}{cos^2(x)}$

$= 1$

-Dan
June 1st 2008, 07:37 PM
unmcintosh

thanks for the help and everything, but could you or someone else explain how you got from the first step to the second please?
June 1st 2008, 07:46 PM
topsquark

Quote:

Originally Posted by unmcintosh

thanks for the help and everything, but could you or someone else explain how you got from the first step to the second please?

$cos(2x) = 2~cos^2(x) - 1$

We also have that
$cos(2x) = 1 - 2~sin^2(x) = cos^2(x) - sin^2(x)$
as well.

See here.

-Dan
June 1st 2008, 08:18 PM
unmcintosh

thank you. my test is tomorrow so im basically screwed. Also, could somebody help me verify
-2cos^2xtan(-x)=sin2x
please?
June 1st 2008, 10:38 PM
Chris L T521

Quote:

Originally Posted by unmcintosh

thank you. my test is tomorrow so im basically screwed. Also, could somebody help me verify
-2cos^2xtan(-x)=sin2x
please?

$-2\cos^2x\tan(-x)=\sin(2x)$

$\implies -2\cos^2x\frac{\sin(-x)}{\cos(-x)}=\sin(2x)$

$\implies -2\cos^2x\frac{-\sin(x)}{\cos(x)}=\sin(2x)$

$\implies -2\cos(x)(-\sin(x))=\sin(2x)$

$\implies 2\sin(x)\cos(x)=\sin(2x)$

This is the case, for this is a double angle identity.

Hope this helped! :D

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