If \( \cos(\theta + \phi) = \frac{3}{5} \) and \( \sin(\theta - \phi) = \frac{5}{13} \), \( 0<\theta, \phi<\frac{\pi}{4} \), then \( \cot(2\theta) \) has the value:

Question

An industrial robot manufacturing company is tasked to design humanoid robots to be used in warehouses where the robots need to pick items from a stack of shelves. The height of the topmost shelf from the ground is 7 feet. To operate, the robot has to move on a track, running parallel to the stack of shelves. The track is fixed 1 foot away from the base of the stack of shelves. Further, the robot cannot bend its arms by more than 60° from the horizontal plane. If the robot’s arms are attached to its shoulder, what should be the minimum height of the robot from the ground to the shoulder for its arms to reach the topmost shelf?

If \( \cos(\theta + \phi) = \frac{3}{5} \) and \( \sin(\theta - \phi) = \frac{5}{13} \), \( 0<\theta, \phi<\frac{\pi}{4} \), then \( \cot(2\theta) \) has the value:

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