Question

How do you solve $\cos \left(x\right)=\sin \left(x\right)$?

Answer 1

Use trigonometry to solve the equation.

Given an equation, $\cos \left(x\right)=\sin \left(x\right)$

Dividing both side by $\cos \left(x\right)$,

$\tan \left(x\right)=1$,

Take inverse on both sides.

$$\begin{gathered} \Rightarrow x={\tan }^{-1}\left(1\right) \\ \Rightarrow x=\frac{\pi }{4} \end{gathered}$$

The period of the ${\displaystyle \tan \left(x\right)}$ function is ${\displaystyle \pi }$ so values will repeat every${\displaystyle \pi }$ radians in both directions.

Therefore, $\mathit{x}\mathbf{=}\mathit{n}\mathit{\pi }\mathbf{+}\frac{\mathbf{\pi }}{\mathbf{4}}$is the solution of the equation $\mathbf{cos}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{sin}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$.