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Problems PreCal 1508 PLTL Workshop, November 16, 2011\\
PLs: Alex Knaust, Edith Mejia. Lecturer: Yi-Yu Liao\\

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Please do the problems that you feel will help your group the most first (you don't have to do them in order). All handouts are available at \url{http://alex.knaust.info/pltlfall2011/}

\begin{enumerate}
\item Timothy presents the following proof for the identity $\sec{y}\cos{y} = 1$
\[ \sec{y}\cos{y} = 1 \quad\quad\quad\quad\text{Original problem}\]
\[ \Longleftrightarrow \sec{y} = \frac{1}{\cos{y}}\quad\quad\quad\quad \text{Divide both sides by } \cos{y}\]
Since this last statement is a known identity, he concludes that $\sec{y}\cos{y} = 1$.
\begin{enumerate}
\item His Teacher gives him 0 pts for the proof and says it is wrong. Where did little Timmy go wrong?
\item Use this technique to 'prove' something that you know not to be true (i.e. $1 = 2$)
\end{enumerate}

\item Use the Pythagorean theorem to derive the identity $\sin^2{x} + \cos^2{x} = 1$

\item Prove (Verify) the following identities. \textbf{Please be very clear about how you get from one step to the next.}
\begin{enumerate}
\item $\tan{t} \cot{t} = 1$
\item $\frac{\tan^2{\theta}}{\sec{\theta}} = \sin{\theta} \tan{\theta}$
\item $\frac{\cot{x}}{\sec{x}} = \csc{x} - \sin{x}$
\item $\frac{\csc{(-x)}}{\sec{(-x)}} = -\cot{x}$
\item $\frac{1 + \sin\theta}{\cos\theta} + \frac{\cos\theta}{1 + \sin\theta} = 2\sec\theta$
\item $\frac{1}{\cos{x} + 1} + \frac{1}{\cos{x} - 1} = -2\csc{x}\cot{x}$
\item $\sin^2 {\alpha} - \sin^4{\alpha} = \cos^2{\alpha} - \cos^4{\alpha}$
\end{enumerate}

\item Use $\sin^2{x} + \cos^2{x} = 1$ to show $1 + \tan^2{x} = \sec^2{x}$ and $1 + \cot^2{x} = \csc^2{x}$

\item Use other identities on your identity chart to show the following identities
\begin{enumerate}
\item $\frac{\tan{x} + \cot{y}}{\tan{x}\cot{y}} = \tan{y} + \cot{x}$
\item $\sec^2\left(\frac{\pi}{2} - x \right) - 1 = \cot^2{x}$
\end{enumerate}

\end{enumerate} %final end/enumerate
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