abreviatsjw

2021-12-30

We have:

$\{\begin{array}{c}2x=y\mathrm{tan}\theta +\mathrm{sin}\theta \\ 2y=x\mathrm{cot}\theta +\mathrm{cos}\theta \end{array}$

And want to prove${x}^{2}+{y}^{2}=1$

My works:

I multiplied first equation by$\mathrm{cos}\theta$ and second one by $\mathrm{sin}\theta$ and get:

$\{\begin{array}{c}2x\mathrm{cos}\theta =y\mathrm{sin}\theta +\mathrm{sin}\theta \mathrm{cos}\theta \\ 2y\mathrm{sin}\theta =x\mathrm{cos}\theta +\mathrm{sin}\theta \mathrm{cos}\theta \end{array}$

By extracting$\mathrm{sin}\theta \mathrm{cos}\theta$ we get:

$2x\mathrm{cos}\theta -y\mathrm{sin}\theta =2y\mathrm{sin}\theta -x\mathrm{cos}\theta$

$x\mathrm{cos}\theta =y\mathrm{sin}\theta$

And want to prove

My works:

I multiplied first equation by

By extracting

Edward Patten

Beginner2021-12-31Added 38 answers

The system

$\{\begin{array}{c}2x\mathrm{cos}\theta =y\mathrm{sin}\theta +\mathrm{sin}\theta \mathrm{cos}\theta \\ 2y\mathrm{sin}\theta =x\mathrm{cos}\theta +\mathrm{sin}\theta \mathrm{cos}\theta \end{array}$

is a linear system of two (independent) equations in two variables, and it is readly checked that$x=\mathrm{sin}\theta ,\text{}\text{}\text{}y=\mathrm{cos}\theta$ is a solution. Therefore it is the unique solution. Now ${x}^{2}+{y}^{2}=1$ follows.

is a linear system of two (independent) equations in two variables, and it is readly checked that

sirpsta3u

Beginner2022-01-01Added 42 answers

You got $x=y\mathrm{tan}\theta$ . Now substitute back into the original equations to get $y=\mathrm{cos}\theta \text{}\text{and}\text{}x=y\mathrm{tan}\theta =\mathrm{sin}\theta$

Vasquez

Expert2022-01-08Added 669 answers

We have from your last step

from this we get

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