First slide

Trigonometric Identities

Question

$x = \sum_{n = 0}^{\infty} \cos^{2 n} θ, y = \sum_{n = 0}^{\infty} \sin^{2 n} θ, z = \sum_{n = 0}^{\infty} \cos^{2 n} θ \sin^{2 n} θ$ $| \cos θ | < 1, | \sin θ | < 1 then x + y + z$ is equal to

Moderate

A

$x y$
B

$y z$
C

$z x$
D

$x y z$

Solution

$\begin{array}{l} x = \frac{1}{1 - \cos^{2} x} = \frac{1}{\sin^{2} x}, y = \frac{1}{\cos^{2} x} \\ and z = \frac{1}{1 - \sin^{2} x \cos^{2} x} = \frac{1}{1 - \frac{1}{x} \cdot \frac{1}{y}} = \frac{x y}{x y - 1} \\ \Rightarrow z (x y - 1) = x y = x + y \\ \Rightarrow x + y + z = x y z . \end{array}$

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