Video footage, which was passed on to North Yorkshire Police, shows the bird becoming caught in the trap in the early hours of May 2.
Shortly afterwards, an individual with their face obscured is seen entering the trap and appears to deliberately kill the bird before removing the body in a bag.
The trap was located on Howl Dale Moor near Goathland.
North Yorkshire Police Wildlife Crime Officer, Jeremy Walmsley, is urging anyone with information to come forward.
”The goshawk is one of the most protected species of bird in the UK and it is extremely distressing that an individual would choose to kill any bird of prey," he said.
"I appeal to anyone with information about this horrific crime to get in touch with the police and help us to find the person responsible for the death of this magnificent bird.
“We see far too many incidents of birds of prey killed or injured in North Yorkshire and as a police force we are doing all we can to put a stop to this inhumane and callous crime.”
Andy Wilson, Chief Executive of the North York Moors National Park Authority said: “We are deeply saddened to hear about this incident. Goshawks were persecuted to extinction in the UK in the late 19th Century and, despite an improvement in numbers, persecution and habitat loss remain a constant threat to their survival.
“Killing or injuring a bird of prey is illegal, cruel and must be prosecuted wherever possible.
"We are working alongside the police to support them in their investigations and we would strongly urge any witnesses or anyone who has any information to come forward.
"With your help the offender(s) can be brought to justice.”
A cage trap can be used to catch certain species of birds and is designed to trap birds alive and unharmed, in case of any non-target species becoming caught.
Any non-target birds, such as birds of prey, should be released as soon as possible after being caught.
Killing a bird of prey is an offence under the Wildlife and Countryside Act 1981.
If you have any information which could help this investigation, please call 101 quoting reference: 12200073462 or if you wish to remain anonymous contact Crimestoppers on 0800 555 111.