Zachariah Ferrell

2023-03-26

How to verify the identity: $\frac{\mathrm{cos}\left(x\right)-\mathrm{cos}\left(y\right)}{\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)}+\frac{\mathrm{sin}\left(x\right)-\mathrm{sin}\left(y\right)}{\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)}=0$?

Genevieve Lin

Check the identity provided:
Given identity: $\frac{\mathrm{cos}\left(x\right)-\mathrm{cos}\left(y\right)}{\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)}+\frac{\mathrm{sin}\left(x\right)-\mathrm{sin}\left(y\right)}{\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)}=0$.
Consider the expression: $\frac{\mathrm{cos}\left(x\right)-\mathrm{cos}\left(y\right)}{\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)}+\frac{\mathrm{sin}\left(x\right)-\mathrm{sin}\left(y\right)}{\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)}$
$=\frac{\left(\mathrm{cos}\left(x\right)-\mathrm{cos}\left(y\right)\right)\left(\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)\right)+\left(\mathrm{sin}\left(x\right)-\mathrm{sin}\left(y\right)\right)\left(\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)\right)}{\left(\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)\right)\left(\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)\right)}=\frac{\left({\mathrm{cos}}^{2}\left(x\right)-{\mathrm{cos}}^{2}\left(y\right)\right)+\left({\mathrm{sin}}^{2}\left(x\right)-{\mathrm{sin}}^{2}\left(y\right)\right)}{\left(\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)\right)\left(\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)\right)}\left(\because \left(a+b\right)\left(a-b\right)={a}^{2}-{b}^{2}\right)=\frac{\left({\mathrm{sin}}^{2}\left(x\right)+{\mathrm{cos}}^{2}\left(x\right)\right)-\left({\mathrm{sin}}^{2}\left(y\right)+{\mathrm{cos}}^{2}\left(y\right)\right)}{\left(\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)\right)\left(\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)\right)}=\frac{1-1}{\left(\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)\right)\left(\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)\right)}\left(\because {\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =1\right)=\frac{0}{\left(\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)\right)\left(\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)\right)}=0$
Hence, the given identity, $\frac{\mathrm{cos}\left(x\right)-\mathrm{cos}\left(y\right)}{\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)}+\frac{\mathrm{sin}\left(x\right)-\mathrm{sin}\left(y\right)}{\mathrm{cos}\left(x\right)+\mathrm{cos}\left(y\right)}=0$ is proved.

Do you have a similar question?