Question
If $\text{cosec x}+\cot \text{x}=\frac{1}{2},0<\text{x}<\frac{\pi}{2},$ then $\cos\text{x}$ is equal to
- $\frac{5}{3}$
- $\frac{3}{5}$
- $-\frac{3}{5}$
- $-\frac{5}{3}$
Solution:
$2\text{cosec}=\frac{1}{2}+2$
$\Rightarrow 2\text{cosec}\text{ x} = \frac{5}{2}$
$\Rightarrow \text{cosec}\text{ x} =\frac{5}{4}$
$\Rightarrow\frac{1}{\sin\text{x}}=\frac{5}{4}$
$\Rightarrow\sin\text{x}=\frac{4}{5}$
Now,
$0<\theta<\frac{\pi}{2}$$\therefore\cos\theta=\sqrt{1-\sin^2\theta}$
$=\sqrt{1-\Big(\frac{4}{5}\Big)^2}$
$=\frac{3}{5}$
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