<s>[INST] 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.
 [/INST] Consider the right triangle with sides $3,4,5$, whose angle opposite the side $3$ is $9\theta$. Then:
$$\sin 9\theta =\frac35 \Rightarrow 5\sin 9\theta=3; \\
\cos 9\theta =\frac45 \Rightarrow 5\cos 9\theta =4;\\
3\csc 3\theta - 4\sec 3\theta=\frac{3}{\sin 3\theta}-\frac4{\cos 3\theta}=\frac{3\cot 3\theta-4\sin 3\theta}{\sin 3\theta \cos 3\theta}=\\
\frac{5\sin 9\theta \cos3\theta-5\cos 9\theta \sin 3\theta}{\sin 3\theta \cos 3\theta}=\\
\frac{5\sin (9\theta -3\theta)}{0.5\sin 6\theta}=10.$$
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