Grove and Benson: FInite Reflection Groups in the context of developing the basic theory of finite reflection groups give the following problem in Chapter 2

Exercise 2.1

Verify that

$\displaystyle x_1 = ( cos \ \theta / 2 , sin \ \theta /2). $

$\displaystyle x_2 = ( -sin \ \theta / 2 , cos \ \theta /2). $

are eigenvectors with respective eigenvalues 1 and -1 for the matrix

$\displaystyle B = \left(\begin{array}{cc}{cos \ \theta}&{sine \ \theta}\\{sin \ \theta}&{- cos \ \theta}\end{array}\right ) $

I would appreciate help with this problem.

Peter