Answer
The solution is $3$.
Work Step by Step
Convert $\frac{\pi }{4}$ into degrees.
$\begin{align}
& \frac{\pi }{4}=\frac{\pi }{4}\left( \frac{180\text{ }\!\!{}^\circ\!\!\text{ }}{\pi } \right) \\
& =\left( \frac{\pi }{\pi } \right)\left( \frac{180\text{ }\!\!{}^\circ\!\!\text{ }}{4} \right) \\
& =45\text{ }\!\!{}^\circ\!\!\text{ }
\end{align}$
Convert $\frac{\pi }{6}$ into degrees.
$\begin{align}
& \frac{\pi }{6}=\frac{\pi }{6}\left( \frac{180\text{ }\!\!{}^\circ\!\!\text{ }}{\pi } \right) \\
& =\left( \frac{\pi }{\pi } \right)\left( \frac{180\text{ }\!\!{}^\circ\!\!\text{ }}{6} \right) \\
& =30\text{ }\!\!{}^\circ\!\!\text{ }
\end{align}$
The expressions for $\tan \theta $ and $\csc \theta $ are
$\tan \theta =\frac{a}{b}$ …… (I)
$\csc \theta =\frac{c}{a}$ …… (2)
Here, $a$ is the length of the side opposite to $\theta $ , $b$ is the length of the side adjacent to $\theta $ and $c$ is the hypotenuse.
In triangle 1,
For $\theta =45\text{ }\!\!{}^\circ\!\!\text{ }$ , $a=1$ and $b=1$
Substitute $\frac{\pi }{4}$ for $\theta $ , $1$ for $a$ and $1$ for $b$ in equation (I).
$\tan \frac{\pi }{4}=\frac{1}{1}$ …… (3)
In triangle 2,
For $\theta =30\text{ }\!\!{}^\circ\!\!\text{ }$ , $a=1$ and $c=2$
Substitute $\frac{\pi }{6}$ for $\theta $ , $1$ for $a$ and $2$ for $c$ in equation (2).
$\csc \frac{\pi }{6}=\frac{2}{1}$ …… (4)
Add equation (4) and equation (3).
$\begin{align}
& \tan \frac{\pi }{4}+\csc \frac{\pi }{6}=\frac{1}{1}+\frac{2}{1} \\
& =3
\end{align}$
Therefore, the solution is $3$.
