Question

If $\tan \theta =p-\frac{1}{4p}$, then $\sec \theta -\tan \theta =$

• A. $2p$ or $\frac{1}{2p}$
• B. $\frac{1}{2p}$ or $-2p$
• C. $-\frac{1}{2p}$ or $2p$
• D. $-\frac{1}{2p}$ or $-2p$

Answer 1

Answer: (B)

$\frac{1}{2p}$ or $-2p$

Given that:

$\tan \theta =p-\frac{1}{4p}$

Let, $\sec \theta -\tan \theta =x\dots \left(1\right)$

$\sec \theta +\tan \theta =\frac{1}{\sec \theta -\tan \theta }$

then, $\sec \theta +\tan \theta =\frac{1}{x}\dots \left(2\right)$

Subtracting $\left(1\right)$ from $\left(2\right)$, we get:

$2\tan \theta =\frac{1}{x}-x=2p-\frac{1}{2p}$

We solve to get $x=-2p$ or $\frac{1}{2p}$

$⟹\sec \theta -\tan \theta =-2p$ or $\frac{1}{2p}$