A simple lemma in trigonometry.

Sunday 12 Jul 2020 in Lemma | 1 min read

Trigonometry Induction

Let \(\theta_1, \theta_2,\ldots, \theta_k \) be angles and \(s_1,\ldots,s_k\) boolean variables in \(\lbrace 0,1\rbrace \). Let \(b_i = \theta_i + s_i \frac{\pi}{2} \). Then \[ \sum_{s_1,\ldots,s_k \in \lbrace 0,1\rbrace^k} \left(\prod_{i=1}^k \cos b_i \right) \cos{\left(\sum_{i=1}^k b_i \right)} = 1. \]

Proof.

By induction. Let \( L_k \) be the expression in Lemma at \( k \). At \(k=1\) \[ L_1 = \sum_{s \in \lbrace 0,1\rbrace} \cos^2\left(\theta +s \frac{\pi}{2}\right) =\cos^2 \theta +\sin^2 \theta = 1. \]

Let us assume that Lemma holds at \(k \) and show that it holds at \(k+1\). Calling \( \varphi_k = \sum_i^k b_i\), we have \(\begin{align*} L_{k+1} & = \sum_{s_1,\ldots,s_k \in \lbrace 0,1\rbrace^k} \left(\prod_{i=1}^{k} \cos b_i \right)\left[\cos{\theta_{k+1}}\cos{\left(\theta_{k+1}+\varphi_k\right)}-\sin{\theta_{k+1}}\cos{\left(\theta_{k+1}+\varphi_k+\frac{\pi}{2}\right)} \right] \\ & = \sum_{s_1,\ldots,s_k \in \lbrace 0,1\rbrace^k} \prod_{i=1}^{k} \cos b_i \left[\cos^2 \theta_{k+1}\cos{\varphi_k} + \sin^2 \theta_{k+1}\cos{\varphi_k} \right] \\ & = \sum_{s_1,\ldots,s_k \in \lbrace 0,1\rbrace^k} \left(\prod_{i=1}^{k} \cos b_i\right) \cos \varphi_k = L_k. \qquad \blacksquare \end{align*}\)