Pythagorean Identities

Provided that all trigonometric functions below are defined for $x$, the three pythagorean identities are:

1. $\sin^{2}(x) + \cos^{2}(x) = 1$
2. $\tan^{2}(x) + 1 = \sec^{2}(x)$
3. $1 + \cot^{2}(x) = \csc^{2}(x)$.

Sum and Difference Formulas

Recall the sum and difference formulas for sine and cosine:

1. $\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)$
2. $\sin(x-y) = \sin(x)\cos(y) - \cos(x)\sin(y)$
3. $\cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y)$
4. $\cos(x-y) = \cos(x)\cos(y) + \sin(x)\sin(y)$.

Double Angle Identities

If we substitute $y=x$ into the addition formulas above, we obtain the double angle formulas for sine and cosine. These formulas are very important in calculus.

1. $\sin(2x) = 2\sin(x)\cos(x)$
2. $\cos(2x) = \cos^{2}(x) - \sin^{2}(x) = 1 - 2\sin^{2}(x) = 2\cos^{2}(x) -1$
3. $\cos^2 x= \displaystyle \frac{1+\cos(2x)}{2}$
4. $\sin^2 x=\displaystyle \frac{1-\cos(2x)}{2}$