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Question Number 203935 by mathlove last updated on 02/Feb/24

$$\frac{cos\frac{x}{3}+sin\frac{x}{3}}{cos\frac{x}{3}-sin\frac{x}{3}}=?$$

Answered by esmaeil last updated on 02/Feb/24

$$=\frac{\sqrt{2}sin(\frac{x}{3}+\frac{\pi }{4})}{\sqrt{2}cos(\frac{x}{3}+\frac{\pi }{4})}=tan(\frac{\pi }{4}+\frac{x}{3})$$

Answered by deleteduser1 last updated on 02/Feb/24

$$Letx=3\theta \Rightarrow \frac{cos\left(\theta \right)+sin\left(\theta \right)}{cos\left(\theta \right)-sin\left(\theta \right)}=\frac{\sqrt{2}sin(\theta +45°)}{\sqrt{2}cos(\theta +45°)}$$$$=tan(\theta +45°)=tan(\frac{x}{3}+\frac{\pi }{4})=tan\left(\frac{4x+3\pi }{12}\right)$$

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