Question

[Key Concept: Unit circle with the given tangent/positive-number setup — use the tangent-length geometry.]

Let the equation of tangent is

$$xcos\theta + ysin\theta = 1$$

( $x_{1}, - 1$ ) lies on tangent

$$\Rightarrow x_{1}cos\theta - sin\theta = 1$$

$$\begin{array}{r} x_{1}cos\theta = 1 + sin\theta\#(1) \end{array}$$

now ( $x_{2},1$ ) lies on tangent

$$\begin{array}{r} \Rightarrow \ x_{2}cos\theta = 1 - sin\theta\#(2) \end{array}$$

(1) $\times (2),\ x_{1}x_{2}\cos^{2}\theta = 1 - \sin^{2}\theta = \cos^{2}\theta$

$$\Rightarrow \ x_{1}x_{2} = 1$$