To start evaluating the given expression, first, replace the pi value with 180 degrees.
[tex]\tan (-\frac{2\pi}{3})=\tan (-\frac{2\times180}{3})=\tan (-\frac{360}{3})=\tan (-120)[/tex][tex]\sin \frac{7\pi}{4}=\sin (\frac{7\times180}{4})=\sin \frac{1260}{4}=\sin 315[/tex][tex]\sec (-\pi)=\sec (-180)=\frac{1}{\cos (-180)}[/tex]
Now that we have converted the radian angles in degrees, let's get the numerical value of each function using calculator.
[tex]\begin{gathered} \tan (-120)=\sqrt[]{3} \\ \sin 315=-\frac{\sqrt[]{2}}{2} \\ \frac{1}{\cos (-180)}=-1 \end{gathered}[/tex]
From that, we can say that the given expression is equal to:
[tex]\begin{gathered} \frac{\sqrt[]{3}}{-\frac{\sqrt[]{2}}{2}}-(-1) \\ =(\frac{\sqrt[]{3}}{1}\times-\frac{2}{\sqrt[]{2}})+1 \\ =\frac{-2\sqrt[]{3}}{\sqrt[]{2}}+1 \\ \text{Rationalize.} \\ =(\frac{-2\sqrt[]{3}}{\sqrt[]{2}}\times\frac{\sqrt[]{2}}{\sqrt[]{2}})+1 \\ =-\sqrt[]{6}+1^{}^{} \end{gathered}[/tex]
The given expression is equal to -√6 + 1 or 1 - √6.
