Hi

Here is a simple way but I do not know if it is in the spirit of the exercise

x² + y² + z² = 1 is the sphere (center O, radius 1)

A plane tangent to the sphere at point M(a,b,c) is perpendicular to vector OM whose coordinates are (a,b,c)

Therefore an equation of the plane is ax + by + cz + k = 0

To find k you just have to substitute (a,b,c) to (x,y,z), M being on the plane

a² + b² + c² + k = 0 therefore k = - a² - b² - c²

An equation of the plane is ax + by + cz - a² - b² - c² = 0