If 3 (x-15^ )= (x+15^ ),0<x<90^ , find x. - Vidyadip

Question

If $3\tan(\text{x}-15^{\circ})=\tan(\text{x}+15^{\circ}),0<\text{x}<90^{\circ},$ find x.

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Answer

Given: $3\tan(\text{x}-15^{\circ})=\tan(\text{x}+15^{\circ})$ $\Rightarrow\frac{\tan(\text{x}+15^{\circ})}{\tan(\text{x}-15^{\circ})}=3$ Applying componendo and dividendo, we have $\frac{\tan(\text{x}+15^{\circ})+\tan(\text{x}-15^{\circ})}{\tan(\text{x}+15)-\tan(\text{x}-15^{\circ})}=\frac{3+1}{3-1}$ $\Rightarrow\frac{\frac{\sin(\text{x}+15^{\circ})}{\cos(\text{x}+15^{\circ})}+\frac{\sin(\text{x}-15^{\circ})}{\cos(\text{x}-15^{\circ})}}{\frac{\sin(\text{x}+15^{\circ})}{\cos(\text{x}+15^{\circ})}-\frac{\sin(\text{x}-15^{\circ})}{\cos(\text{x}-15^{\circ})}}=\frac{4}{2}$ $\Rightarrow\frac{\sin(\text{x}+15^{\circ})\cos(\text{x}-15^{\circ})+\cos(\text{x}+15^{\circ})\sin(\text{x}-15^{\circ})}{\sin(\text{x}+15^{\circ})\cos(\text{x}-15^{\circ})-\cos(\text{x}+15^{\circ})\sin(\text{x}-15^{\circ})}=2$ $\Rightarrow\frac{\sin(\text{x}+15^{\circ}+\text{x}-15^{\circ})}{\sin(\text{x}+15^{\circ}-\text{x}+15^{\circ})}=2$ $\Rightarrow\frac{\sin2\text{x}}{\sin30^{\circ}}=2$ $\Rightarrow\sin2\text{x}=2\times\frac{1}{2}=1$ $\Big(\sin30^{\circ}=\frac{1}{2}\Big)$ $\Rightarrow\sin2\text{x}=\sin90^{\circ}$ $\Rightarrow2\text{x}=90^{\circ}$ $(0<\text{x}<90^{\circ})$ $\Rightarrow\text{x}=45^{\circ}$

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