If cosec θ=1312, find the value of 2 sin θ−3 cos θ4 sin θ−9 cos θ.
Question:

If $\operatorname{cosec} \theta=\frac{13}{12}$, find the value of $\frac{2 \sin \theta-3 \cos \theta}{4 \sin \theta-9 \cos \theta}$.

Solution:

Given: $\operatorname{cosec} \theta=\frac{13}{12}$

We have to find the value of the expression $\frac{2 \sin \theta-3 \cos \theta}{4 \sin \theta-9 \cos \theta}$.

Now,

$\operatorname{cosec} \theta=\frac{13}{12}$

$\Rightarrow \sin \theta=\frac{1}{\operatorname{cosec} \theta}=\frac{1}{\frac{13}{12}}=\frac{12}{13}$

$\cos \theta=\sqrt{1-\sin ^{2} \theta}=\sqrt{1-\left(\frac{12}{13}\right)^{2}}=\frac{5}{13}$

Therefore,

$\frac{2 \sin \theta-3 \cos \theta}{4 \sin \theta-9 \cos \theta}=\frac{2 \times \frac{12}{13}-3 \times \frac{5}{13}}{4 \times \frac{12}{13}-9 \times \frac{5}{13}}$

$=3$

Hence, the value of the expression is 3.