The couple then posed for pictures with the stunning cliff top monastery in the background.
The couple chatted briefly with a guide, Kate asking: 'Do you have many rescues here?'
But they declined offers of water or to sit down on two a specially created seats draped with fabric under a parasol.
'Half-way, let's keep going, let's keep moving,' he said.
Before they moved on the couple, guided by the director of the national museum, Phuntesho Tashi, span a series of prayer wheels in the picturesque spot, which in Buddhism helps to absolve sin.
The prince looked slightly more flushed than his wife and wiped sweat from his brow.
'So far so good,' he laughed. 'This is absolutely stunning.'
The couple stopped to talk to three trekkers from San Francisco who were returning down from the mountain, Lauren McKennan, 29, Chris Steele, 27, and Alex Willmore, 28.
'They were pretty altogether,' said Lauren. 'There wasn't a single trail of sweat. They stopped to talk to us for three to five minutes and were absolutely charming. '
Alex added: 'They asked us how long we had been here and talked about how lovely the weather was. They said how rainy it had been when they were doing the archery and how glad they were doing the trek today.
'We asked for a photograph they just laughed and wouldn't do it. But they were absolutely charming, very cool, very nice. They kept on saying how beautiful it was.'
The monastery itself dates back to 1692, although the sacred and rather mystical nature of the spot is even older.
Legend has it that Guru Padmasambhava, who is credited with bringing Buddhism to Bhutan, flew to the site on a tigresses back to subdue a local demon in the 8thCentury.
The monastery is sited near a cave where Guru Padmasambhava subsequently meditated in for three years, three months, three weeks and three days.
Taktshang Goemba, or Tiger's Nest, was later blessed and sanctified as one of Bhutan's most sacred religious sites.
An aide said: 'It is a magical place that the Duke and Duchess are thrilled to be visiting. Their hike will take between five and six hours in total and will allow them to get a real sense of the natural and spiritual beauty of the country.'