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This is the grey gear on the Home Screen. If you have a Verizon iPhone, you won't be able to use your phone's settings to disable caller ID. It's in the same section of apps as Messages and FaceTime. If your carrier doesn't support this feature (as in the case of Verizon phones), you won't see this option here. It will turn white. From now on, the people you call won't be able to see your phone number.
Open your iPhone's Settings. Scroll down and tap Phone. Tap Show My Caller ID. Slide the Show My Caller ID switch left.