I think it would be a good idea to review some important formulas from algebra and trigonometry before we get into serious stuff in calculus.

**Expansion of Polynomials**

- \((a+b)^2=a^2+2ab+b^2\)
- \((a-b)^2=a^2-2ab+b^2\)
- \((a+b)^3=a^3+3a^2b+3ab^2+b^3\)
- \((a-b)^3=a^3-3a^2b+3ab^2-b^3\)

**Factorization of Polynomials**

- \(a^2-b^2=(a+b)(a-b)\)
- \(a^3-b^3=(a-b)(a^2+ab+b^2)\)
- \(a^3+b^3=(a+b)(a^2-ab+b^2)\)

**Trigonometric Identities**

- \(\cos^2\theta+\sin^2\theta=1\)
- \(\tan^2\theta+1=\sec^2\theta\)

**Sine Sum and Difference Formulas**

- \(\sin(\theta_1+\theta_2)=\sin\theta_1\cos\theta_2+\cos\theta_1\sin\theta_2\)
- \(\sin(\theta_1-\theta_2)=\sin\theta_1\cos\theta_2-\cos\theta_1\sin\theta_2\)

**Sine Double Angle Formula** \[\sin2\theta=2\sin\theta\cos\theta\]

**Cosine Sum and Difference Formulas**

- \(\cos(\theta_1+\theta_2)=\cos\theta_1\cos\theta_2-\sin\theta_1\sin\theta_2\)
- \(\cos(\theta_1-\theta_2)=\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2\)

**Cosine Double Angle Formula** \begin{eqnarray*}\cos2\theta&=&\cos^2\theta-\sin^2\theta\\&=&2\cos^2\theta-1\\&=&1-2\sin^2\theta\end{eqnarray*}

From this Cosine Double Angle Formula, we obtain Half Angle Formulas.

**Half Angle Formulas**

- \(\cos^2\theta=\displaystyle\frac{1+\cos\theta}{2}\) or equivalently \(\cos\theta=\pm\sqrt{\displaystyle\frac{1+\cos\theta}{2}}\)
- \(\sin^2\theta=\displaystyle\frac{1-\cos\theta}{2}\) or equivalently \(\sin\theta=\pm\sqrt{\displaystyle\frac{1-\cos\theta}{2}}\)