This Destiny: The Taken King feature will tell you how to beat The Sunbreaker’s Challenge quest for your Titan, which grants access to the Hammer of Sol. Before you can take part, you need to make sure you beat the Coming War, which is the first story mission you'll encounter in The Taken King expansion.
After completing said mission, head over to the Tower and have a conversation with Commander Zavala. He'll provide the Sunbreaker's Challenge to you, although you must complete this mission solo. No fire team this time.
Make your way to your ship and take a look at the planet Venus. You should see a quest beacon flashing there. Take it, and the Ghost will help with navigation as you go through the stage. You'll come across Dig Site 4, which you might have visited before when you were asked to battle the Nexus. You'll come across a number of Vex, which you should be able to take care of. The Ultra Hydra will pose a challenge afterwards, so take it down and then Scan the Conflux with your Ghost. That will end the current stage.
Head back to Orbit, then look for another mission on Venus called Bound By Fire. You'll see a PvP map that may be familiar to you, depending on how much you played Destiny. After landing, go ahead and find the center room located at the far end of the map. You will rely upon your Ghost twice while in this area.
Once that's done, the Sunbreaker subclass will be yours, and you'll be able to put the newfound ability, the Hammer of Sol, to good use. It recharges quickly (well, for this particular mission anyway), so go nuts with throwing hammers at whatever foes come your way. The general purpose here is to get an idea of how it works, and how powerful it can be against certain foes. Once you've cleared the room of Vex, you'll conclude this part of the current mission.
Once this is done, go back to Orbit and return to the Tower. You'll run into Zavala again, who will provide you a Sunbreaker's Mark at 220 Defense. Although you can't use it right away due to the lacking Light level, you will be able to infuse it.
Your current quest will come to a close. If you're using a Hunter Nightstalker or Warlock Stormcaller subclass, there will be quests that work on a similar one as this. They're worth completing, since they will provide you with a trio of formidable Guardians to use throughout The Taken King. If you need further assistance bumping up your level, consult our Level 40 Guide.
Make sure you also check with our Destiny: The Taken King Walkthrough and Guide for more tips.