Solve the following equations: + 2x= 3x - Vidyadip

Question

Solve the following equations: $\tan\text{x}+\tan2\text{x}=\tan3\text{x}$

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Answer

$\tan\text{x}+\tan2\text{x}=\tan(\text{x}+2\text{x})$ $\tan\text{x}+\tan2\text{x}-\frac{\tan\text{x}+\tan2\text{x}}{1-\tan\text{x}\tan2\text{x}}=0$ $[\tan\text{x}+\tan2\text{x}]\Big[1-\frac{1}{1-\tan\text{x}\tan2\text{x}}\Big]=0$ $[\tan\text{x}+\tan2\text{x}]\Big[\frac{1-\tan\text{x}\tan2\text{x}-1}{1-\tan\text{x}\tan2\text{x}}\Big]=0$ $[\tan\text{x}+\tan2\text{x}]\Big[\frac{-\tan\text{x}\tan2\text{x}}{1-\tan\text{x}\tan2\text{x}}\Big]=0$ $\tan\text{x}=0$ or $\tan2\text{x}=0$ or $\tan\text{x}+\tan2\text{x}=0$ $\text{x}=\text{n}\pi$ or $\frac{\text{n}\pi}{2}$ or $\tan\text{x}\Big[\frac{1-\tan^{2}\text{x}+2}{1-\tan^{2}\text{x}}\Big]=0$ $\text{x}=\text{n}\pi$ or $\frac{\text{n}\pi}{2}$ or $\tan\text{x}=\pm\sqrt{3}$$\text{x}=\text{m}\pi$ or $\frac{\text{n}\pi}{3}\text{m,n}\in\text{Z}$

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