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# Fundamental Trigonometric Identities

These are your basic building blocks for solving trigonometric equations and understanding how the pieces fit together. Using these identities can make sense of even the scariest looking trig.

Evaluate the sum

\[\begin{align} & \ \ \ \log_{\cos1}(\tan{1}) \\ &+\log_{\cos{2}}(\tan{2}) \\ &+ \log_{\cos{3}}(\tan{3}) \\& +\ldots \\ &+ \log_{\cos{44}} (\tan{44}) \\ &+ \log_{\sin{45}}(\tan{45}) \\ &+ \log_{\sin46}(\tan{46}) \\ &+\ldots \\ &+ \log_{\sin89}(\tan{89}). \end{align}\]

**Note:** All angles are in degrees, and be aware that the base changes from \( \cos \) to \( \sin\).

\[\large (7\cos x+24\sin x)(7\sin x-24\cos x) \]

Find the maximum value of this expression over all real values \(x.\)

Hint:

If \(x\) and \(y\) are acute angles such that

\[ \frac {\sin x}{\sin y } = \frac {1}{2}, \quad \frac {\cos x}{\cos y } = \frac 3 2 , \]

what is \( \tan^2 (x+y) \)?

Define the function \(f(x)=\frac{2x}{1-x^2}\). Find the number of distinct **real** solutions of the equation \(f^{(5)} (x) =x.\)

**Details and assumptions**

\( f^{(n)} (x) \) denotes the function \(f\) applied \(n\) times. In particular, \( f^{(5)} (x) = f(f(f(f(f(x)))))\).

Find \[\cos{1˚} \cos{2˚}+\cos{2˚} \cos{3˚}+ \cdots +\cos{88˚} \cos{89˚}.\]

Give your answer to two decimal places.

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