If θ is positive acute angle and 3(sec²θ + tan²θ) = 5, then the value of cos2θ is?

Solution (By Examveda Team)

$$\eqalign{ & {\text{3}}\left( {{{\sec }^2}\theta + {\text{ta}}{{\text{n}}^2}\theta } \right) = 5 \cr & {\sec ^2}\theta + {\text{ta}}{{\text{n}}^2}\theta = \frac{5}{3}\,.....(i) \cr & {\sec ^2}\theta - {\text{ta}}{{\text{n}}^2}\theta = 1\,......(ii) \cr & {\text{Adding equation (i) and (ii)}} \cr & {\text{2}}{\sec ^2}\theta = \frac{8}{3} \cr & sec\theta = \frac{2}{{\sqrt 3 }} \cr & \therefore \theta = {30^ \circ } \cr & \cos 2\theta = \cos 2\left( {{{30}^ \circ }} \right) \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \cos {60^ \circ } \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{1}{2} \cr} $$