Question

If $tan\theta =m-\frac{1}{4m}$, then find k such that $sec\theta -tan\theta =-\frac{1}{km}$

Answer 1

given $sec\left(\theta \right)-tan\left(\theta \right)=-\frac{1}{km}$ let it be eqn 1

and $tan\left(\theta \right)=m-\frac{1}{4m}$

we know $se{c}^2\left(\theta \right)-ta{n}^2\left(\theta \right)=1$

$\Rightarrow sec\left(\theta \right)+tan\left(\theta \right)=\frac{1}{sec\left(\theta \right)-tan\left(\theta \right)}=-km$ let it be eqn 2

now subtracting 2 from 1 we get $2tan\left(\theta \right)=-km+\frac{1}{km}$

$\Rightarrow tan\left(\theta \right)=\frac{-km}{2}+\frac{1}{2km}$

comparing with given equation we get

$-\frac{k}{2}=1$ so $k=-2$