If is a positive acute angle and 2 . 3 = 1 then the value of ( 2co s^2 5 2 - 1 ) is?

Answer: Option C

$$\begin{array}{rl}& \tan 2\theta .\tan 3\theta =1\\ & \left(2\theta +3\theta \right)=90^{\circ }\\ & 5\theta =90^{\circ }\\ & \left[\text{If tan A}\text{.tan B}=\text{1}\right]\\ & \left[\text{then, A}+\text{B}={90}^{\circ }\right]\\ & \Rightarrow \left(\text{2co}{\text{s}}^2\frac{5\theta }{2}-1\right)\\ & \Rightarrow \text{2co}{\text{s}}^2\frac{{90}^{\circ }}{2}-1\\ & \Rightarrow \text{2co}{\text{s}}^{2}45^{\circ }-1\\ & \Rightarrow \frac{\text{2}}{2}-1\\ & \Rightarrow 0\end{array}$$