Very-Short and Short-Answer Questions: If =x, write the value of … - Vidyadip

Question

Very-Short and Short-Answer Questions:
If $\sec\theta=\text{x},$ write the value of $\tan\theta.$

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Answer

$\sec\theta=\text{x}$
$\Rightarrow\frac{1}{\cos\theta}=\text{x}$
$\Rightarrow\cos\theta=\frac{1}{\text{x}}$
$\Rightarrow\cos^2\theta=\frac{1}{\text{x}^2}$
$\Rightarrow\sin^2\theta=1-\cos^2\theta=1-\frac{1}{\text{x}^2}=\frac{\text{x}^2-1}{\text{x}}$
$\Rightarrow\sin\theta=\frac{\sqrt{\text{x}^2-1}}{\text{x}}$
Now, $\tan\theta=\frac{\sin\theta}{\cos\theta}$
$=\frac{\frac{\sqrt{\text{x}^2-1}}{\text{x}}}{\frac{1}{\text{x}}}=\sqrt{\text{x}^2-1}$

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