Given:
[tex]\tan \frac{7\pi}{8}[/tex]
To Determine: The identity that is equivalent to the given tangent
Note that, the identity rule below would be applied
[tex]\tan \frac{\alpha}{2}=\sqrt[]{\frac{1-\cos \alpha}{1+\cos \alpha}}[/tex]
Also,
[tex]\tan \frac{\alpha}{2}=\frac{\sin \alpha}{1+\cos \alpha}[/tex]
And also,
[tex]\tan \frac{\alpha}{2}=\frac{1-\cos \alpha}{\sin \alpha}[/tex]
From the given tangent, we can re-write it as below:
[tex]\begin{gathered} \tan \frac{7\pi}{8}\cong\tan \frac{\frac{7\pi}{4}}{2} \\ \text{Note} \\ \frac{7\pi}{8}=\frac{\frac{7\pi}{4}}{2} \end{gathered}[/tex]
Therefore:
[tex]\tan \frac{\frac{7\pi}{4}}{2}=\sqrt[]{\frac{1-\cos\frac{7\pi}{4}}{1+\cos\frac{7\pi}{4}}}[/tex]
Also:
[tex]\tan \frac{\frac{7\pi}{4}}{2}=\frac{\sin \frac{7\pi}{4}}{1+\cos \frac{7\pi}{4}}[/tex]
And also,
[tex]\tan \frac{\frac{7\pi}{4}}{2}=\frac{1-\cos \frac{7\pi}{4}}{\sin \frac{7\pi}{4}}[/tex]
It can be observed from the option provided, the correct options is
I and III only
