Climate is much more than just latitude
Things like altitude, geography, air currents, water currents, all of that factors in.
England for example is at the same latitude as parts of Canada that are covered in snow much of the year, yet England has a much milder climate. This is because the north Atlantic ocean current drives warm water and subsequently warm air right to the British isles.
As for the Sahara it is believed that northern Africa regularly went through periods of humidity where it rained a lot and there were lakes and rivers and forests full of life, and periods where the air was dry and all the water dried up and everything died. This was caused by small changes in the planets tilt on its axis. Then at some point some factor changed in this sensitive balance that threw the whole system out of whack.
Current thinking is that this coincides with the spread of humans and agriculture across africa. As northern africa was transitioning from a humid period to a dry period humans transitioned from hunter-gatherer to domesticated animals and farming in order to survive which could have lead to over grazing, over farming and destruction of vegetation. This vegetation would have been needed to provide the moisture in the air to trigger a new humid phase when the time came. Without it the next humid phase was never triggered and the system spiralled out of control leading to the eventual creation of the Sahara desert.
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