# Trigonometric identity help

• Mar 31st 2010, 07:54 PM
Bikkstah
Trigonometric identity help
Hi all,

I am stuck on proving these two problems:

(csc theta - sin theta)^2 = cot^2theta - cos^2theta

and

(1+cos2theta/cos^2theta)=2

I know I need to use double angle identities in the second one, but I have been stuck on how to get going.
• Mar 31st 2010, 07:58 PM
Prove It
Quote:

Originally Posted by Bikkstah
Hi all,

I am stuck on proving these two problems:

(csc theta - sin theta)^2 = cot^2theta - cos^2theta

and

(1+cos2theta/cos^2theta)=2

I know I need to use double angle identities in the second one, but I have been stuck on how to get going.

$\displaystyle (\csc{\theta} - \sin{\theta})^2 = \csc^2{\theta} - 2\csc{\theta}\sin{\theta} + \sin^2{\theta}$

$\displaystyle =\csc^2{\theta} - 2 + \sin^2{\theta}$

$\displaystyle = \csc^2{\theta} - 1 - (1 - \sin^2{\theta})$

$\displaystyle = \cot^2{\theta} - \cos^2{\theta}$.
• Apr 1st 2010, 12:57 PM
Bikkstah
Can anyone help with (1 + Cos2theta/cos^2theta)=2? I know it's true by setting theta equal to a real angle like 30 degrees, but I don't know how to prove it using just identities.
• Apr 1st 2010, 01:03 PM
e^(i*pi)
Quote:

Originally Posted by Bikkstah
(1+cos2theta/cos^2theta)=2

Use the identity $\displaystyle \cos (2\theta) = 2\cos^2 \theta -1$

Spoiler:
$\displaystyle \frac{1+\cos (2\theta)}{\cos^2 \theta}$

$\displaystyle \frac{1+2\cos^2 \theta -1}{\cos^2 \theta}$

This cancels down to 2
• Apr 1st 2010, 02:35 PM
Bikkstah
Thank you both so much!