# Prove the following

• Nov 23rd 2013, 12:21 AM
anil86
Prove the following
• Nov 23rd 2013, 07:09 PM
chiro
Re: Prove the following
Hey anil86.

I'd suggest you take a look at the sum-to-product and product-to-sum identities (and keeping in mind the relation cos(ix) and cosh(x) and so on).

List of trigonometric identities - Wikipedia, the free encyclopedia
• Nov 24th 2013, 02:47 AM
Idea
Re: Prove the following
Restating the problem:Given
$\displaystyle \sin (\theta +i \varphi )=\tan \alpha + i \sec \alpha$
or
$\displaystyle \sin \theta \cosh \varphi = \tan \alpha$
$\displaystyle \cos \theta \sinh \varphi = \sec \alpha$
Show
$\displaystyle \cos 2\theta \cosh 2\varphi = 3$

Let
$\displaystyle a=\cos ^2\theta \text{ }$ and $\displaystyle b=\sin ^2\theta$

$\displaystyle c=\cosh ^2\varphi$ and $\displaystyle d=\sinh ^2\varphi$

so that

$\displaystyle a+b=1$

$\displaystyle c-d=1$

$\displaystyle a d - b c = 1$

$\displaystyle a-b=\cos 2\theta$

$\displaystyle c+d= \cosh 2\varphi$

Now use the identity

$\displaystyle (a-b)(c+d)=2(a d-b c)+(a+b)(c-d)$