Solve the following equations: cosec x=1+ - Vidyadip

Question

Solve the following equations:
$\text{cosec }\text{x}=1+\cot\text{x}$

✓

Answer

We have,
$\text{cosec }\text{x}=1+\cot\text{x}$
$\Rightarrow\frac{1}{\sin\text{x}}=1+\frac{\cos\text{x}}{\sin\text{x}}$
$\Rightarrow1=\sin\text{x}+\cos\text{x}$
Divide both side by $\sqrt{2}$, We get,
$\Rightarrow\frac{1}{\sqrt2}\sin\text{x}+\frac{1}{\sqrt2}\cos\text{x}=\frac{1}{\sqrt2}$
$\Rightarrow\sin\frac{\pi}{4}\sin\text{x}+\cos\frac{\pi}{4}\cos\text{x}=\frac{1}{\sqrt2}$
$\Rightarrow\cos\Big(\text{x}-\frac{\pi}{4}\Big)=\cos\frac{\pi}{4}$
$\Rightarrow\text{x}=\frac{\pi}{4}=2\text{n}\pi\pm\frac{\pi}{4},\text{n}\in\text{z}$
$\therefore\text{x}\Big(2\text{n}\pi+\frac{\pi}{2}\Big)$ Or $2\text{n }\pi,\text{n}\in\text{z}$

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