If P(x) = [(cosx, sinx),(-sinx, cosx)], then show that P(x).P(y) = P(x + y) = P(y).P(x)

Question

If P(x) = \(\begin{bmatrix} cosx & sinx \\[0.3em] -sinx & cosx \end{bmatrix}\) , then show that P(x).P(y) = P(x + y) = P(y).P(x).

Answer 1

Given,

We know that-

cos x cos y + sin x sin y = cos (x – y)

cos x sin y + sin x cos y = sin (x + y)

and cos x cos y – sin x sin y = cos (x + y)

By matrix multiplication, we have –

∴ From equation 2 and 3:

P(x).P(y) = P(y).P(x) = P(x + y)