Question

If $\sec \theta =\sqrt{2}$ and $\frac{3\pi }{2}<\theta <2\pi$, then find the value of $\frac{1+\tan \theta +\text{cosec}\theta }{1+\cot \theta -\text{cosec}\theta }$.

Answer 1

$\sec \theta =\sqrt{2}$ for $\theta =\frac{7\pi }{4}$ in the range of $\theta$ given in the question.

In IV quadrant, $\tan$ and $\text{cosec}$, both are -ve.

$\Rightarrow \tan \left(\frac{7\pi }{4}\right)=-1$ and $\text{cosec}\left(\frac{7\pi }{4}\right)=-\sqrt{2}$

Putting these values in the question, we get

$\frac{1-1-\sqrt{2}}{1-1+\sqrt{2}}=-1$