CLP-1 Differential Calculus

\begin{align*} \sin^2\theta + \cos^2 \theta &=1 & \xmapsto{\text{divide by $\cos^2\theta$}}&& \tan^2\theta + 1 &= \sec^2\theta\\ \sin^2\theta + \cos^2 \theta &=1 & \xmapsto{\text{divide by $\sin^2\theta$}}&& 1 + \cot^2 \theta &=\csc^2\theta \end{align*}

\begin{align*} \sin(2\alpha) &= 2\sin(\alpha)\cos(\alpha) \end{align*}

\begin{align*} \cos(2\alpha) &= \cos^2(\alpha) - \sin^2(\alpha)\\ &= 2\cos^2(\alpha) - 1 & \text{(use $\sin^2(\alpha)= 1-\cos^2(\alpha)$)}\\ &= 1 - 2\sin^2(\alpha) & \text{(use $\cos^2(\alpha)= 1-\sin^2(\alpha)$)} \end{align*}

\begin{align*} \cos( \arcsin x) &= \sqrt{1-x^2} & \sec( \arctan x) &= \sqrt{1+x^2} \end{align*}