Rewriting Trigonometric Powers and Products

Definition For every real number $u$,

These are the angle addition formulas with $A = B = u$. For all real $A$ and $B$,

To see why, rotate the plane counterclockwise about the origin by the angle $A$. The rotation sends $(1, 0)$ to $(\cos A, \sin A)$ and $(0, 1)$ to $(-\sin A, \cos A)$, and because it preserves sums and scalar multiples it sends $(p, q) = p(1, 0) + q(0, 1)$ to $(p\cos A - q\sin A,\ p\sin A + q\cos A)$. The point $(\cos B, \sin B)$ at angle $B$ lands at angle $A + B$, which is the point $(\cos(A + B), \sin(A + B))$, and comparing coordinates gives both formulas. The last two forms of $\cos 2u$ come from replacing $\sin^2 u$ or $\cos^2 u$ using $\sin^2 u + \cos^2 u = 1$.

Definition The notation $\sin^2 u$ means $(\sin u)^2$, the square of the output, and likewise for $\cos^2 u$ and other positive integer powers. It does not mean $\sin(u^2)$. By contrast $\sin^{-1} u$ is the inverse sine $\arcsin u$, not a reciprocal, so the superscript convention applies only to positive powers.

Definition Solving the double-angle identities for the squares gives, for every real $u$,

No square root and no sign choice is involved, because these formulas rewrite the squared function itself. A sign question appears only if you try to undo the square, since the root of the right side of the first formula is $\lvert \cos u \rvert$, not $\cos u$.

Definition Every identity above holds for any real angle $u$, so $u$ may be replaced by a whole expression such as $3x$, $\frac{x}{2}$ or $x + 1$. The replacement must be made for every occurrence of $u$ at once, keeping the whole expression together as one angle. In particular $2u$ becomes twice the whole expression.

Definition Adding and subtracting the angle addition formulas for $A + B$ and $A - B$ gives, for all real $A$ and $B$,

Definition A purposeful rewrite replaces an expression with an equivalent one chosen for a stated goal. A frequent goal in calculus is a sum of constants and single first powers $\sin kx$ or $\cos kx$. The working rule is to reduce even powers of sine and cosine with power reduction, to split products of different angles with product-to-sum, and to stop as soon as the target form is reached.

Definition Two expressions in $x$ are equivalent when they are defined for the same inputs and give equal values at every one of them. An identity proves equivalence. Checking a few numerical values can catch a mistake, but agreement at finitely many points is evidence, not proof.