Question

If A(α) = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]` then prove that A²(α) = A(2α)

Answer 1

A²(α) = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)] [(cos alpha, sin alpha),(-sin alpha, cos alpha)]`

= `[(cos^2 alpha - sin^2 alpha, cos alpha sin alpha + sin alpha cos alpha),(- cos alpha sin alpha - sin alpha cos  alpha, - sin^2 alpha cos^2 alpha)]`

= `[(cos(alpha + alpha), sin(alpha + alpha)),(- sin(alpha + alpha), cos(alpha + alpha))]`    .......`[(∵ sin("A" + "B") = sin "A" cos "B" + cos "A" sin "B"),(cos("A" + "B") = cos "A" cos "B" - sin "A" sin "B")]`

= `[(cos(2alpha), sin(2alpha)),(-sin(2alpha), cos(2alpha))]`

= A(2α)