If $\tan 9\theta = 3/4$, then find the value of $3\csc 3\theta - 4\sec 3\theta$. 
If $\tan9\theta=\dfrac{3}{4}$, where $0<\theta<\dfrac{\pi}{18}$, then find the value of $3\csc 3\theta - 4\sec 3\theta$.

My approach:-
$$\begin{align*} \tan9\theta &=\frac{3}{4} \\[6pt] \implies \theta & = \frac{37^{\circ}}{3} \end{align*}$$
By using this, we get value of  $(3\csc3\theta - 4\sec3\theta) =9.95$ by using calculator. 
I want know if there's any way to solve this problem without calculator.

Hint: We get $$\theta=\frac{1}{9}\arctan(\frac{3}{4})$$ so we get
$$3\csc\left(\frac{1}{3}\arctan(\frac{3}{4})\right)-4\sec\left(\frac{1}{3}\arctan(\frac{3}{4})\right)=10$$
