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Introduction of Trigonometry — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsIntroduction of Trigonometry1 Mark
MCQ
If $\sec\theta+\tan\theta=\text{x},$ then $\sec\theta=$
  • A
    $\frac{\text{x}^2+1}{\text{x}}$
  • ✓
    $\frac{\text{x}^2+1}{2\text{x}}$
  • C
    $\frac{\text{x}^2-1}{2\text{x}}$
  • D
    $\frac{\text{x}^2-1}{\text{x}}$
✓

Answer

Correct option: B.
$\frac{\text{x}^2+1}{2\text{x}}$
Given, $\sec\theta+\tan\theta=\text{x}$
We know that,
$\sec^2\theta-\tan^2\theta=1$
$\Rightarrow (\sec\theta+\tan\theta)(\sec\theta-\tan\theta)=1$
$\Rightarrow\ \text{x}(\sec\theta-\tan\theta)=\frac{1}{\text{x}}$
Now,
$\sec\theta+\tan\theta=\text{x},$
$\sec\theta-\tan\theta=\frac{1}{\text{x}}$
Adding the two equations, we get
$(\sec\theta+\tan\theta)+(\sec\theta-\tan\theta)=\text{x}+\frac{1}{\text{x}}$
$\Rightarrow\ \sec\theta+\tan\theta+\sec\theta-\tan\theta=\frac{\text{x}^2+1}{\text{x}}$
$\Rightarrow\ 2\sec\theta=\frac{\text{x}^2+1}{\text{x}}$
$\Rightarrow\ \sec\theta=\frac{\text{x}^2+1}{2\text{x}}$
Therefore, the correct choice is $(b).$

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