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# Pythagorean Identities

Trigonometric identities bring new life to the Pythagorean theorem by re-envisioning the legs of a right triangle as sine and cosine.

# Pythagorean Identities: Level 4 Challenges

For all real values of $$\theta$$ for which $$\lvert \sin \theta \rvert \neq 1,$$ evaluate:

$\large \sum_{n = 1}^{\infty} \sin^{2n}\theta .$

Find the value of $E=\tan^2{\dfrac{\pi}{40}}+\tan^2{\dfrac{3\pi}{40}}+\tan^2{\dfrac{5\pi}{40}}+\cdots+\tan^2{\dfrac{19\pi}{40}}.$

$\begin{cases} \cos \alpha = \tan \beta \\ \cos \beta = \tan \gamma \\ \cos \gamma = \tan \alpha \end{cases}$

The acute angles $$\alpha, \beta$$ and $$\gamma$$ satisfy the system of equations above. Find $$\sin \gamma$$.

$\begin{cases} \dfrac{\sin\theta}{x} =\dfrac{\cos\theta}{y} \\ \dfrac{\cos^{4}\theta}{x^{4}}+\dfrac{\sin^{4}\theta}{y^{4}}=\dfrac{97\sin 2\theta}{x^{3}y+y^{3}x} \end{cases}$

Let $$x$$ and $$y$$ be positive real numbers and $$\theta$$ is an angle such that it is not a multiple of $$\frac{\pi}{2}$$. If $$x,y$$ and $$\theta$$ satisfy the system of equations above, find $$\dfrac{x}{y}+\dfrac{y}{x}$$.

###### Source: 2009 Harvard-MIT Mathematics Tournament

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 $$\tan^2 (x+y)$$?

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