If sec + tan sec - tan = 2 5179 , then the value of is?

Answer: Option C

$$\begin{array}{rl}& \text{Given, }\frac{\text{sec}\theta +\text{tan}\theta }{\text{sec}\theta -\text{tan}\theta }=\text{2}\frac{51}{79}\\ & \Rightarrow \frac{\text{sec}\theta +\text{tan}\theta }{\text{sec}\theta -\text{tan}\theta }=\frac{209}{79}\\ & \text{By componendo dividendo}\\ & \left[\frac{a}{b}=\frac{c}{d},\frac{a+b}{a-b}=\frac{c+d}{c-d}\right]\\ & \Rightarrow \frac{\text{sec}\theta +\text{tan}\theta +\text{sec}\theta -\text{tan}\theta }{\text{sec}\theta +\text{tan}\theta -\text{sec}\theta +\text{tan}\theta }=\frac{209+79}{209-79}\\ & \Rightarrow \frac{2sec\theta }{2\text{tan}\theta }=\frac{288}{130}\\ & \Rightarrow \frac{sec\theta }{\text{tan}\theta }=\frac{288}{130}\\ & \Rightarrow \frac{\frac{1}{\text{cos}\theta }}{\frac{\sin \theta }{\text{cos}\theta }}=\frac{288}{130}\\ & \Rightarrow \frac{1}{\sin \theta }=\frac{288}{130}\\ & \Rightarrow \text{Therefore, }\sin \theta =\frac{130}{288}\\ & \Rightarrow \sin \theta =\frac{65}{144}\end{array}$$