Follow these steps to derive the law of cosines.

✓ 1. The relationship between the side lengths in △ABD is
c² = x² + h² by the cosine relationship law of sines Pythagorean theorem .

✓ 2. The relationship between the side lengths in △CBD is
a² = (b − x)² + h² by the cosine relationship law of sines Pythagorean theorem .

✓ 3. The equation a² = (b − x)² + h² is expanded factored reduced to become
a² = b² − 2bx + x² + h².

✓ 4. Using the equation from step 1, the equation
a² = b² − 2bx + x² + h² becomes a² = b² − 2bx + c²
by substitution symmetry .

✓ 5. In △ABD, the trigonometric function cos(A) cos(B) cos(C) = #&\frac {x}{c}#&.

✓ 6. Multiply both sides of the equation in step 5 by a b c
to get x = c cos(A).

✓ 7. Substitute x/c c cos(A) h for the variable x in the equation
a² = b² − 2bx + c² to produce a² = b² − 2bc cos(A) + c².

✓ 8. The law of cosines is obtained by rearranging solving the previous equation.
a² = b² + c² − 2bc cos(A)