Question
Very-Short and Short-Answer Questions.If $5\text{x}=\sec\theta$ and $\frac{5}{\text{x}}=\tan\theta,$ find the value of $5\Big(\text{x}^2-\frac{1}{\text{x}^2}\Big).$
✓
Answer
Given, $5\text{x}=\sec\theta$ and $\frac{5}{\text{x}}=\tan\theta$
We know that,
$1+\tan^2\theta=\sec^2\theta$
$\Rightarrow1+\Big(\frac{5}{\text{x}}\Big)^2=(5\text{x})^2$
$\Rightarrow25\text{x}^2-\frac{25}{\text{x}^2}=1$
$\Rightarrow25\Big(\text{x}^2-\frac{1}{\text{x}^2}\Big)=1$
$\Rightarrow5\Big(\text{x}^2-\frac{1}{\text{x}^2}\Big)=\frac15$
We know that,
$1+\tan^2\theta=\sec^2\theta$
$\Rightarrow1+\Big(\frac{5}{\text{x}}\Big)^2=(5\text{x})^2$
$\Rightarrow25\text{x}^2-\frac{25}{\text{x}^2}=1$
$\Rightarrow25\Big(\text{x}^2-\frac{1}{\text{x}^2}\Big)=1$
$\Rightarrow5\Big(\text{x}^2-\frac{1}{\text{x}^2}\Big)=\frac15$
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