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Answer:
[tex]\displaystyle \frac{\sin \theta + \sec \theta}{\sin \theta - \sec\theta} = -3[/tex]
Step-by-step explanation:
We are given that:
[tex]\displaystyle \frac{\sin \theta}{\sec \theta} = \frac{1}{2}[/tex]
And we want to find the value of:
[tex]\displaystyle \frac{\sin \theta + \sec \theta}{\sin \theta - \sec\theta}[/tex]
From the second expression, we can divide both the numerator and denominator by sec(θ). Thus:
[tex]\displaystyle = \frac{ \dfrac{\sin \theta + \sec \theta}{\sec \theta} }{ \dfrac{\sin \theta - \sec\theta}{\sec \theta} }[/tex]
Simplify:
[tex]\displaystyle = \frac{\dfrac{\sin \theta}{\sec \theta} + 1}{\dfrac{\sin\theta}{\sec\theta} - 1}[/tex]
Since we know that sin(θ) / sec(θ) = 1 / 2:
[tex]\displaystyle = \frac{\left(\dfrac{1}{2}\right)+1}{\left(\dfrac{1}{2}\right)-1}[/tex]
Evaluate:
[tex]\displaystyle = \frac{\dfrac{3}{2}}{-\dfrac{1}{2}} = -3[/tex]
Therefore:
[tex]\displaystyle \frac{\sin \theta + \sec \theta}{\sin \theta - \sec\theta} = -3[/tex]