If =x+1 4x , then + = - Vidyadip

MCQ

If $\tan\text{x}=\text{x}+\frac{1}{4\text{x}},$ then $\sec\text{x}+\tan\text{x}=$

✓
$-2\text{x},\frac{1}{2\text{x}}$
B
$-\frac{1}{2\text{x}},2\text{x}$
C
$2\text{x}$
D
$2\text{x},\frac{1}{\text{x}2}$

✓

Answer

Correct option: A.

$-2\text{x},\frac{1}{2\text{x}}$

We have:
$\tan\text{x}=\text{x}-\frac{1}{4\text{x}}$
$\Rightarrow\sec^2\text{x}=1+\tan^2\text{x}$
$\Rightarrow\sec^2\text{x}=1+\Big(\text{x}-\frac{1}{4\text{x}}\Big)^2$
$\Rightarrow\sec^2\text{x}=\text{x}^2+\frac{1}{16\text{x}^2}+\frac{1}{2}$
$\Rightarrow \sec\text{x}^2 =\Big(\text{x}+\frac{1}{4\text{x}}\Big)^2$
$\therefore \sec\text{x}=\pm\Big(\text{x}+\frac{1}{4\text{x}}\Big)$
$\Rightarrow \sec\text{x}-\tan\text{x} =\Big(\text{x}+\frac{1}{4\text{x}}\Big)- \Big(\text{x}-\frac{1}{4\text{x}}\Big)\text{ or} -\Big(\text{x}+\frac{1}{4\text{x}}\Big)\text{or}- \Big(\text{x}-\frac{1}{4\text{x}}\Big)$
$=\frac{1}{2\text{x}}$ or $-2\text{x}$

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