Question

how does the auxin promote phototropism

Answer 1

Auxin increases the flexibility of plant cell walls which are the main factor limiting plant cell growth. As a consequence, plant cells tend to grow faster in the presence of auxin. When a plant senses a directional light stimulus it redistributes its auxin molecules to the side that is further away from the light. As a consequence, the plant cells that are further away from the light experience a greater auxin concentration and grow faster, causing the shoot to bend towards the light. This mechanism allows plants to optimize the amount of light they receive in order to increase photosynthesis rates. 

Answer 2

when the sunlight come near to plant perpendicular then auxin hormones in the tip of stem spread uniformly. so the stem grow equally rapidly to the above side due two equal presence of auxin both side of stem. when the light Falls slightly on the plant auxin hormones once collect in another side of stem so bend toward the sunlight because auxin hormones prefer to stay in shade.
hope this helps you.