If $$\frac{\text{sec}\text{θ+\text{tan}\text{θ}}}{{\text{sec}\text{θ-\text{tan}\text{θ}}}}=\frac{5}{3}$$​, then the value of sin θ is:

Correct option is D

Given:

​$$\frac{\sec\theta + \tan\theta}{\sec\theta - \tan\theta} = \frac{5}{3}$$

Formula Used:​

sec$$\theta =\frac 1{\cos \theta}$$ , tan $$\theta =\frac{\sin\theta} {\cos \theta}$$​​

Solution:
$$\frac{\sec\theta + \tan\theta}{\sec\theta - \tan\theta} = \frac{5}{3}$$

​​$$\frac{\frac 1{\cos\theta} + \frac{\sin\theta}{\cos\theta}}{\frac 1{\cos\theta} -\frac{\sin\theta}{\cos\theta}} = \frac{5}{3}$$​​

​$$\frac{1 + \sin\theta}{1 - \sin\theta} = \frac{5}{3}$$​​

​$$3(1 + \sin\theta) = 5(1 - \sin\theta)$$​​

​$$3 + 3\sin\theta = 5 - 5\sin\theta$$​

​$$3\sin\theta + 5\sin\theta = 5 - 3 = 2$$​

8 $$\sin\theta = 2$$​

​$$\sin\theta = \frac{1}{4}$$​