Let O be the center of the circle ACBD.

Fact-1: All the same color coded angles are equal.

Fact-2: The red dotted circle will pass through A,P, B, O.

Fact-3: $\angle purple1 = \angle green + \angle red$, a standard result from tangent properties.

By considering $\triangle ADQ$, $\angle blue = \angle green + \angle red$.

Since $\angle purple1 = \angle blue$, we say that A, P, B, Q, O are con-cyclic.

Then, $\angle black = \angle purple2 = \angle purple1 = \angle black$

That means $\angle green + \angle red = \angle green + \angle yellow$.

Result follows.

Let O be the center of the circle ACBD.

Fact-1: All the same color coded angles are equal.

Fact-2: The red dotted circle will pass through A,P, B, O.

Fact-3: $\angle purple1 = \angle green + \angle red$, a standard result from tangent properties.

By considering $\triangle ADQ$, $\angle blue = \angle green + \angle red$.

Since $\angle purple1 = \angle blue$, we say that A, P, B, Q, O are con-cyclic.

Then, $\angle black = \angle purple2 = \angle purple1$

That means $\angle green + \angle yellow = \angle green + \angle red$.

Result follows.