Given the expression:
[tex]\frac{\tan(-\frac{2\pi}{3})}{\sin(\frac{7\pi}{4})}-\sec (-\pi)[/tex]
We will find each trigonometric function the substitute into the expression
We will find each value with the help of the reference angle
[tex]\tan (-\frac{2\pi}{3})=\tan (\frac{4\pi}{3})=\tan (\pi+\frac{\pi}{3})=\tan (\frac{\pi}{3})=\sqrt[]{3}[/tex][tex]\sin (\frac{7\pi}{4})=\sin (2\pi-\frac{\pi}{4})=-\sin (\frac{\pi}{4})=-\frac{1}{\sqrt[]{2}}[/tex][tex]\sec (-\pi)=\sec (-\pi+2\pi)=\sec \pi=-1[/tex]
Substitute with the recent values:
[tex]\begin{gathered} \frac{\tan(-\frac{2\pi}{3})}{\sin(\frac{7\pi}{4})}-\sec (-\pi)=(\sqrt[]{3}\div-\frac{1}{\sqrt[]{2}})-(-1) \\ \\ =(\sqrt[]{3}\cdot(-\sqrt[]{2}))+1=-\sqrt[]{6}+1 \end{gathered}[/tex]
So, the answer will be:
[tex]-\sqrt[]{6}+1[/tex]
