# trig identities

• Aug 9th 2010, 02:53 PM
JCM133
trig identities
I’m stuck on the following problem (see Attachment) any help would be greatly appreciated

Attachment 18487

I’m not sure how it goes from here to the next line
Thanks
Kind regards
• Aug 9th 2010, 03:31 PM
skeeter
you need to review your post ...
• Aug 9th 2010, 04:00 PM
JCM133
Trigonometric Identities
Hi All
Im having a problem with the problem in the attached file

Attachment 18488

any help would be greatly appreciated
Thanks Kind Regards

John
• Aug 9th 2010, 05:19 PM
eumyang
$2 \left( \sin \frac{\theta}{2} \cos \frac{\theta}{2} \cos^2 \frac{\phi}{2} + \sin \frac{\phi}{2} \cos \frac{\phi}{2} \cos^2 \frac{\theta}{2} + \sin \frac{\phi}{2} \cos \frac{\phi}{2} \sin^2 \frac{\theta}{2} + \sin \frac{\theta}{2} \cos \frac{\theta}{2} \sin^2 \frac{\phi}{2}\right)$
(I rearranged the factors in the last term to make it consistent.)

Distribute the 2:
$2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \cos^2 \frac{\phi}{2} + 2 \sin \frac{\phi}{2} \cos \frac{\phi}{2} \cos^2 \frac{\theta}{2} + 2 \sin \frac{\phi}{2} \cos \frac{\phi}{2} \sin^2 \frac{\theta}{2} + 2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \sin^2 \frac{\phi}{2}$

Note that each term begins with $2\sin \frac{\theta}{2} \cos \frac{\theta}{2}$ or $2\sin \frac{\phi}{2} \cos \frac{\phi}{2}$. Recall the sine of a double angle identity: $\sin 2u = 2\sin u \cos u$ So
$2\sin \frac{\theta}{2} \cos \frac{\theta}{2} = \sin \left( 2\cdot \frac{\theta}{2} \right) = \sin \theta$, and the same for angle phi.

Replace:
$\sin \theta \cos^2 \frac{\phi}{2} + \sin \phi \cos^2 \frac{\theta}{2} + \sin \phi \sin^2 \frac{\theta}{2} + \sin \theta \sin^2 \frac{\phi}{2}$

Rearrange:
$\sin \theta \cos^2 \frac{\phi}{2} + \sin \theta \sin^2 \frac{\phi}{2} + \sin \phi \cos^2 \frac{\theta}{2} + \sin \phi \sin^2 \frac{\theta}{2}$

Factor:
$\sin \theta \left( \cos^2 \frac{\phi}{2} + \sin^2 \frac{\phi}{2} \right) + \sin \phi \left( \cos^2 \frac{\theta}{2} + \sin^2 \frac{\theta}{2} \right)$

Use the Pythagorean identity:
$\sin \theta (1) + \sin \phi (1)$

$\sin \theta + \sin \phi$
• Aug 10th 2010, 12:14 AM
JCM133
Thanks EUMYANG that has helped a bundle

kind regards
John