Solve the following equations: 3 + =5 cosec x - Vidyadip

Question

Solve the following equations:
$3\tan\text{x}+\cot\text{x}=5\ \text{cosec }\text{x}$

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Answer

$3\tan\text{x}+\cot\text{x}=5\ \text{cosec}\ \text{x}$ $\Rightarrow\frac{3\sin\text{x}}{\cos\text{x}}+\frac{\cos\text{x}}{\sin\text{x}}=\frac{5}{\sin\text{x}}$ $\Rightarrow\frac{3\sin^2+\cos^2\text{x}}{\cos\text{x}\sin\text{x}}=\frac{5}{\sin\text{x}}$ $\Rightarrow3(1-\cos^2\text{x})+\cos^2\text{x}=5\cos\text{x}$ $\Rightarrow3-3\cos^2\text{x}+\cos^2\text{x}=5\cos\text{x}$ $\Rightarrow2\cos^2\text{x}+5\cos\text{x}-3=0$ $\Rightarrow2\cos^2\text{x}+6\cos\text{x}-\cos\text{x}-3=0$ $\Rightarrow2\cos\text{x}(\cos\text{x}+3)-1(\cos\text{x}+3)=0$ $\Rightarrow(2\cos\text{x}-1)(\cos\text{x}+3)=0$ $\Rightarrow(2\cos\text{x}-1)=0$ or $\cos\text{x}+3=0$ $\Rightarrow\cos\text{x}=\frac{1}{2}$ or $\cos\text{x}=-3$$\cos\text{x}=-3$ is not possible $(\therefore-1\leq\cos\text{x}\leq1)$
$\Rightarrow\cos\text{x}=\cos\frac{\pi}{3}$
$\Rightarrow\text{x}=2\text{n}\pi\pm\frac{\pi}{3},\ \text{n}\in\text{Z}$

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