Geometry
# Fundamental Trigonometric Identities

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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