How to memorize trigonometric identities without rote memorization learn the connections

Heres a systematic approach that reduces memorization to just THREE core identities: **Core identities (memorize these):** 1. $\sin^2 \theta + \cos^2 \theta = 1$ 2. $\sin(A+B) = \sin A \cos B + \cos A \sin B$ 3. $\cos(A+B) = \cos A \cos B - \sin A \sin B$ **Everything else comes from these:** **Double angle:** Set $B = A$: $$\sin 2A = 2 \sin A \cos A$$ $$\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$$ **Half angle:** Solve $\cos 2\theta = 1 - 2\sin^2 \theta$ for $\sin\theta$: $$\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}$$ **Using Eulers formula** ($e^{i\theta} = \cos\theta + i\sin\theta$) is the most elegant approach: $$e^{i(A+B)} = e^{iA}e^{iB}$$ $$\cos(A+B) + i\sin(A+B) = (\cos A + i\sin A)(\cos B + i\sin B)$$ Multiply out and equate real/imaginary parts — you get both addition formulas instantly! **For product-to-sum:** Use the addition formulas with $P = (A+B)/2$ and $Q = (A-B)/2$. The $\pm$ signs in half-angle formulas are determined by which quadrant $\theta/2$ lies in. On a unit circle, you can derive this visually.