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\begin{center}
\textbf{Trigonometric Identities - March 13, 2002}
\end{center}

Prove all the trigonometric identities.

\begin{enumerate}

\item  $\tan x + 1 = (\sec x)(\sin x + \cos x)$
	\\Hint: convert to only sines and cosines.

% not true:
\item $\sin x + \cos x = \tan x$
	\\Hint: do you believe this identity?


\item  
	\[
	\frac{\cos \theta + \sin \theta}{\cos \theta} = 1+ \tan \theta
	\]
	\\ Hint: Can you use mostly algebra?

\item  \[
	\sec 2y = \frac{\sec^2y}{2-\sec^2y}
	\]
	\\ Hint: convert to only sines and cosines.

\item  \[
	\frac{\cos^3 \alpha - \sin^3 \alpha}{\cos \alpha - \sin \alpha} = \frac{2+\sin 2 \alpha}{2}
	\]
	\\ Hint: do you remember how to factor a difference of cubes?  ($x^3-y^3=?$)

\item  $2\cos A \cos B = \cos(A-B) + \cos(A+B)$
\item  
	\[
	\frac{\cos x + \sin x}{\cos x - \sin x} - \frac{\cos x - \sin x}{\cos x + \sin x} = 2\tan 2x
	\]
	\\ Hint: common denominators for LHS.  What can you do to the right side?


\item  $1-\cos 2\theta + \cos 4\theta - \cos 6\theta = 4\sin \theta \cos 2\theta \sin 3\theta$
	\\ Hint: First simplify the following: $\cos 2\theta$, $\cos 4\theta$, $\cos 6\theta$, $\sin 3\theta$.

\item $\tan \theta + \tan(\theta+120^\circ) + \tan(\theta+240^\circ) = 3 \tan 3\theta$.
	\\ Hint: What can you do with both sides?


\end{enumerate}









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