Question Number 203935 by mathlove last updated on 02/Feb/24
$$\frac{{cos}\frac{{x}}{\mathrm{3}}+{sin}\:\frac{{x}}{\mathrm{3}}}{{cos}\:\frac{{x}}{\mathrm{3}}−{sin}\:\frac{{x}}{\mathrm{3}}}=? \\ $$
Answered by esmaeil last updated on 02/Feb/24
$$=\frac{\sqrt{\mathrm{2}}{sin}\left(\frac{{x}}{\mathrm{3}}+\frac{\pi}{\mathrm{4}}\right)}{\:\sqrt{\mathrm{2}}{cos}\left(\frac{{x}}{\mathrm{3}}+\frac{\pi}{\mathrm{4}}\right)}={tan}\left(\frac{\pi}{\mathrm{4}}+\frac{{x}}{\mathrm{3}}\right) \\ $$
Answered by AST last updated on 02/Feb/24
$${Let}\:{x}=\mathrm{3}\theta\Rightarrow\frac{{cos}\left(\theta\right)+{sin}\left(\theta\right)}{{cos}\left(\theta\right)−{sin}\left(\theta\right)}=\frac{\sqrt{\mathrm{2}}{sin}\left(\theta+\mathrm{45}°\right)}{\:\sqrt{\mathrm{2}}{cos}\left(\theta+\mathrm{45}°\right)} \\ $$$$={tan}\left(\theta+\mathrm{45}°\right)={tan}\left(\frac{{x}}{\mathrm{3}}+\frac{\pi}{\mathrm{4}}\right)={tan}\left(\frac{\mathrm{4}{x}+\mathrm{3}\pi}{\mathrm{12}}\right) \\ $$