Trigonometric identity help

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March 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.
March 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.

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

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

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

$= \cot^2{\theta} - \cos^2{\theta}$ .
April 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.
April 1st 2010, 01:03 PM
e^(i*pi)

Quote:

Originally Posted by Bikkstah

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

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

Spoiler:

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

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

This cancels down to 2
April 1st 2010, 02:35 PM
Bikkstah

Thank you both so much!