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Find all real x such that sin x = cos x

by RoRi

Find all x \in \mathbb{R} such that

    \[ \sin x = \cos x. \]

We set \sin x = \cos x and compute,

    \begin{align*} \cos x = \sin x &&\implies && \cos x - \sin x &= 0 \\ && \implies && \cos^2 x - 2 \sin x \cos x + \sin^2 x &= 0 \\ && \implies && \sin 2x &= \sin^2 + \cos^2 x \\ && \implies && \sin 2x &= 1 \\ && \implies && 2x &= \frac{\pi}{2} + 2n \pi \\ && \implies && x &= \frac{\pi}{4} + n \pi \qquad \text{for } n \in \mathbb{Z}. \end{align*}

In the second to last line we used this exercise in which we found all real x such that \sin x = 0.

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