Question

If $\sin \left(\mathrm{x}\right)+\sin \left(\mathrm{y}\right)+\sin \left(\mathrm{z}\right)=3$ find the value of $\cos \left(\mathrm{x}\right)+\cos \left(\mathrm{y}\right)+\cos \left(\mathrm{z}\right)$.

Answer 1

Compare range of LHS and RHS

$$\begin{gathered} -1\le \sin \left(\mathrm{x}\right)\le 1\forall \mathrm{x}\in \mathrm{ℝ} \\ -1\le \sin \left(\mathrm{y}\right)\le 1\forall \mathrm{x}\in \mathrm{ℝ} \\ -1\le \sin \left(\mathrm{z}\right)\le 1\forall \mathrm{x}\in \mathrm{ℝ} \end{gathered}$$

Adding all the three above Equations

$\Rightarrow -3\le \sin \left(\mathrm{x}\right)+\sin \left(\mathrm{y}\right)+\sin \left(\mathrm{z}\right)\le 3$

Which is possible only when $\sin \left(\mathrm{x}\right)=\sin \left(\mathrm{y}\right)=\sin \left(\mathrm{z}\right)=1$

$\Rightarrow \cos \left(\mathrm{x}\right)=\cos \left(\mathrm{y}\right)=\cos \left(\mathrm{z}\right)=0\left[\because {\sin }^2\left(\mathrm{x}\right)+{\cos }^2\left(\mathrm{x}\right)=1\right]$

Hence the Value of $\cos \left(\mathrm{x}\right)+\cos \left(\mathrm{y}\right)+\cos \left(\mathrm{z}\right)$ is $0$.