MCQ
If $\text{3x}=\text{cosec}\theta$ and $\frac{3}{\text{x}}=\cot\theta$ then $3\Big(\text{x}^2-\frac{1}{\text{x}^2}\Big)=?$
- A$\frac{1}{27}$
- B$\frac{1}{81}$
- ✓$\frac{1}{3}$
- D$\frac{1}{9}$
✓
Answer
Correct option: C.
$\frac{1}{3}$
We know that
$\text{cosec}^2\theta-\cot^2\theta=1$
$\Rightarrow(\text{3x})^2-\Big(\frac{3}{\text{x}}\Big)^2=1$
$\Rightarrow\text{9x}^2-\frac{9}{\text{x}^2}=1$
$\Rightarrow9\Big(\text{x}^2-\frac{1}{\text{x}^2}\Big)=1$
$\Rightarrow\Big(\text{x}^2-\frac{1}{\text{x}^2}\Big)=\frac{1}9{}$
$\Rightarrow3\Big(\text{x}^2-\frac{1}{\text{x}^2}\Big)=\frac{1}3{}$
$\text{cosec}^2\theta-\cot^2\theta=1$
$\Rightarrow(\text{3x})^2-\Big(\frac{3}{\text{x}}\Big)^2=1$
$\Rightarrow\text{9x}^2-\frac{9}{\text{x}^2}=1$
$\Rightarrow9\Big(\text{x}^2-\frac{1}{\text{x}^2}\Big)=1$
$\Rightarrow\Big(\text{x}^2-\frac{1}{\text{x}^2}\Big)=\frac{1}9{}$
$\Rightarrow3\Big(\text{x}^2-\frac{1}{\text{x}^2}\Big)=\frac{1}3{}$
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