Converting between sine and cosine

In today's blog, I show two lemmas that are useful for converting between sin and cosine.

Lemma 1: cos(x) = sin(π/2 - x)

Proof:

(1) cos(x) = sin(x + π/2) [See Lemma 1, here]

(2) sin(x + π/2) = -sin(-x -π/2) [See Property 4, here]

(3) -sin(-x - π/2) = sin(-x - π/2 + π) [See Corollary 1.2, here]

(4) sin(-x -π/2 + π) = sin(π/2 - x)

QED

Corollary 1.1: cos(π/2 - x) = sin(x)

Proof:

(1) cos(x) = sin(π/2 - x) [See Lemma 1 above]

(2) Let x = π/2 - z

(3) cos(π/2 - z) = sin(π/2 - [π/2 - z]) = sin(z)

QED