I highly recommend the hike from Parque Lage to the Corcovado. It is 2.7km (1.7mi) long and one gains about 700m in altitude, so it is rather steep. There is a short section of about 20m where one has to climb up some rocks, holding a chain, and a few other places where on should hold on to roots and trees go get up the hill. Otherwise it is not excessively challenging for a reasonably fit person. (However, I would not bring children on this.)
After a few minutes there was almost no sign of civilization anymore (ignoring Rio's urban white noise and the ever-present helicopters) and I felt like I was in a remote rain forest. Really enjoyable!
It took me just under an hour to get to Corcovado - it's a great way to get up there without spending a long time in traffic or waiting for the tramway. On the way up I met no one. On the way down I saw 4-5 other parties. There was almost no litter, zero mosquitoes, and no other bothersome critters. I saw one monkey. The greatest danger seemed to be the mangoes raining from the trees down in Parque Lage.
Someone has posted a little trail map on the internet. It is not to scale, but I found it helpful, especially to find the trail head in Parque Lage, which is not marked.
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