If $$\sec \theta + \frac{1}{{\cos \theta }} = 2,$$    find the value of $${\sec ^{55}}\theta + \frac{1}{{{{\sec }^{55}}\theta }} = ?$$

Solution (By Examveda Team)

$$\eqalign{ & \sec \theta + \frac{1}{{\cos \theta }} = 2 \cr & \sec \theta + \sec \theta = 2 \cr & \sec \theta = 1 \cr & {\sec ^{55}}\theta + \frac{1}{{{{\sec }^{55}}\theta }} \cr & = {\left( 1 \right)^{55}} + \frac{1}{{{{\left( 1 \right)}^{55}}}} \cr & = 1 + 1 \cr & = 2 \cr} $$