How do you prove that the sum of the angles of any triangle always equals 180 degrees using trigonometric functions and identities?

To prove the sum of the angles of any triangle equals 180 degrees using trigonometric functions and identities, consider a triangle with angles A, B, and C. Using the identity for the tangent of the sum of two angles, tan(A + B) = (tan A + tan B) / (1 – tan A tan B). Since tan(C) = tan(180° – (A + B)) and tan(180° – x) = -tan(x), it follows that tan(A + B) = -tan(C). This implies that A + B + C = 180°.

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