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Use an original charger and a lightning cable
If the Touch ID doesn't work when charging, it is possible that you are charging your device with a cheap, fake charger, which may prevent the Touch ID sensor from functioning properly.
Test if your device's Touch ID will work, when the device is being charged using an original Apple charger. If it does, replace the fake charger with an original one, purchased from an authorized Apple store.
If your MacBook, iPad or iPhone touch ID doesn't respond, it is possible that the Touch ID sensor is dirty and it can't recognize your fingerprint. Grab a piece of soft cloth, gently clean the sensor and try again.
Maybe your device's Touch ID works, but it "forgot" your fingerprint pattern. In this case, consider deleting your fingerprint pattern and adding it again. If you are not sure how, follow the steps below:
Click on the Apple menu icon in the upper left corner of the screen and select System Preferences;
Click on Touch ID;
Move the mouse pointer over the fingerprint icon you want to delete and click on X in the upper left corner, once it appears;
Enter your password to confirm and click on Delete;
Repeat steps 1 and 2, click on Add a fingerprint and follow the on-screen instructions.
Maybe the Touch ID works perfectly but it doesn't recognize your fingerprint because you are using the wrong finger. Every finger on our hands has a different fingerprint pattern so, if you used your thumb when setting up the Touch ID, the Touch ID sensor will recognize only the thumb.
Note: If you want, you may save up to 5 fingerprint patterns in your device, in Settings -> Touch ID & Passcode.
Maybe the Touch ID doesn't work because your device "forgot" your fingerprint pattern. In order to fix this and make the Touch ID sensor recognize your fingerprint pattern again, you will have to "retrain" it.
To retrain your device, follow the steps below:
Go to Settings -> Touch ID & Passcode;
Enter your passcode, tap on a fingerprint pattern and tap on Delete Fingerprint; *Delete all the fingerprint patterns if you have more than one, to make way for new ones;
Tap on Add a Fingerprint... and follow the on-screen instructions;
Feel free to watch the video below if you are not entirely sure on how to add fingerprint patterns:
Note: Ensure to wipe the Touch ID sensor before "training" it.
Maybe the Touch ID sensor has malfunctioned and is unable to recognize your nor any other person's fingerprint patterns. If restoring the device didn't work, a hardware-related issue is definitely the reason and in his case you should take your device to an authorised repair shop.
If you tried all the other solutions to make the Touch ID work, but unsuccessfully, consider restoring your device, via iTunes or via the device itself. Before restoring your device, make sure to make a backup of all your data to avoid losing them.
Maybe the Touch ID doesn't work because the repair guy forgot to attach the Touch ID sensor to the motherboard, or, in a worst-case scenario, they have damaged the flex cable of the sensor. To be on the safe side, take your device to the repair shop, to have this issue resolved.