If $$\theta $$ is a positive acute angle and $$\tan 2\theta .\tan 3\theta $$    = 1 then the value of $$\left( {{\text{2co}}{{\text{s}}^2}\frac{{5\theta }}{2} - 1} \right)$$   is?

Solution(By Examveda Team)

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