Answer
The required value of $\tan 112.5{}^\circ $ is $-\sqrt{2}-1$.
Work Step by Step
We have to find the value of $\tan 112.5{}^\circ $; the formula is used that represents the tan identity in terms of cos and sine. And the term $\tan 112.5{}^\circ $ comes under the second quadrant where the value of sine and cos functions is positive and the rest of the trigonometric functions are negative. The value of the second quadrant is between $90{}^\circ \,\text{ and }\,18\text{0}{}^\circ $.
$\begin{align}
& \tan 112.5{}^\circ =\tan \frac{225{}^\circ }{2} \\
& =\frac{1-\cos 225{}^\circ }{\sin 225{}^\circ } \\
& =\frac{1-\left( -\frac{\sqrt{2}}{2} \right)}{-\frac{\sqrt{2}}{2}} \\
& =\frac{2+\sqrt{2}}{-\sqrt{2}}
\end{align}$
The denominator is taken with the individual numerator in order to solve the equation.
$\begin{align}
& \tan 112.5{}^\circ =-\frac{2}{\sqrt{2}}-1 \\
& =-\sqrt{2}-1
\end{align}$
Hence, the value of $\tan 112.5{}^\circ $ is $-\sqrt{2}-1$.
