Fundamental Trigonometric Identities

For every real $x$, $\sin ^2x+\cos ^2x=1$, $\sin(-x)=-\sin x$, and $\cos(-x)=\cos x$. Wherever the displayed quotients are defined, $\tan(-x)=-\tan x$, $\cot(-x)=-\cot x$, $\sec(-x)=\sec x$, $\csc(-x)=-\csc x$, $1+\tan^2x=\sec^2x$, and $1+\cot^2x=\csc^2x$. The conventions and prerequisite facts used below are recorded in Parity and the Pythagorean identity for sine and cosine, Tangent, cotangent, secant, and cosecant on their exact natural domains.

For every real $x$, $\sin(\pi/2-x)=\cos x,\quad \cos(\pi/2-x)=\sin x,\quad \sin(\pi-x)=\sin x,\quad \cos(\pi-x)=-\cos x,$ and the quarter-turn and reflection formulas are $\sin(x+\pi/2)=\cos x,\quad \cos(x+\pi/2)=-\sin x,\quad \sin(-x)=-\sin x,\quad \cos(-x)=\cos x.$ On the common natural domains of the two sides, $\tan(\pi/2-x)=\cot x,\quad \cot(\pi/2-x)=\tan x,\quad \sec(\pi/2-x)=\csc x,\quad \csc(\pi/2-x)=\sec x,$ and $\tan(\pi-x)=-\tan x,\quad \cot(\pi-x)=-\cot x,\quad \sec(\pi-x)=-\sec x,\quad \csc(\pi-x)=\csc x.$ The conventions and prerequisite facts used below are recorded in Pythagorean and parity identities for all six trigonometric functions on their natural domains, Quarter-turn values and shifts by pi/2 and pi, The addition formulas for sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi, Tangent, cotangent, secant, and cosecant on their exact natural domains.

For all real $u,v$, $\sin(u-v)=\sin u\cos v-\cos u\sin v,\qquad \cos(u-v)=\cos u\cos v+\sin u\sin v.$ The conventions and prerequisite facts used below are recorded in The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine.

If $\cos u\cos v\cos(u+v)\ne0$, then $\tan(u+v)=\frac{\tan u+\tan v}{1-\tan u\tan v},\qquad \sec(u+v)=\frac{\sec u\sec v}{1-\tan u\tan v};$ if $\cos u\cos v\cos(u-v)\ne0$, then $\tan(u-v)=\frac{\tan u-\tan v}{1+\tan u\tan v},\qquad \sec(u-v)=\frac{\sec u\sec v}{1+\tan u\tan v}.$ If $\sin u\sin v\sin(u+v)\ne0$, then $\cot(u+v)=\frac{\cot u\cot v-1}{\cot u+\cot v},\qquad \csc(u+v)=\frac{\csc u\csc v}{\cot u+\cot v};$ if $\sin u\sin v\sin(u-v)\ne0$, then $\cot(u-v)=\frac{\cot u\cot v+1}{\cot v-\cot u},\qquad \csc(u-v)=\frac{\csc u\csc v}{\cot v-\cot u}.$ The conventions and prerequisite facts used below are recorded in Tangent, cotangent, secant, and cosecant on their exact natural domains, The addition formulas for sine and cosine, The subtraction formulas for sine and cosine.

For every real $x$, $\sin(2x)=2\sin x\cos x,\quad \cos(2x)=\cos^2x-\sin^2x=2\cos^2x-1=1-2\sin^2x,$ and $\sin^2x=(1-\cos2x)/2,\qquad \cos^2x=(1+\cos2x)/2.$ When defined, $\tan(2x)=2\tan x/(1-\tan^2x)$. The conventions and prerequisite facts used below are recorded in The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, Tangent, cotangent, secant, and cosecant on their exact natural domains.

For every real $x$, $\sin3x=3\sin x-4\sin^3x,\qquad \cos3x=4\cos^3x-3\cos x.$ If $\tan x$ and $\tan3x$ are defined and $1-3\tan^2x\ne0$, then $\tan3x=(3\tan x-\tan^3x)/(1-3\tan^2x)$. The conventions and prerequisite facts used below are recorded in Double-angle and quadratic power-reduction identities, The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, Tangent, cotangent, secant, and cosecant on their exact natural domains.

For every real $x$, $\cos(x/2)=\varepsilon_c\sqrt{(1+\cos x)/2},\qquad \sin(x/2)=\varepsilon_s\sqrt{(1-\cos x)/2},$ where $\varepsilon_c,\varepsilon_s\in\{-1,0,1\}$ are respectively the signs of $\cos(x/2)$ and $\sin(x/2)$ (so $\operatorname{sgn}(0)=0$). Thus the positive square root is valid only where the relevant half-angle function is nonnegative. The conventions and prerequisite facts used below are recorded in Double-angle and quadratic power-reduction identities, Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$; the positives are $\{x^2 : x \neq 0\}$, Signs, monotonicity intervals, and ranges of sine and cosine, Tangent, cotangent, secant, and cosecant on their exact natural domains.

For all real $u,v$, $\sin u\sin v=\frac{\cos(u-v)-\cos(u+v)}2,\quad \cos u\cos v=\frac{\cos(u-v)+\cos(u+v)}2,$ $\sin u\cos v=\frac{\sin(u+v)+\sin(u-v)}2,$ and, with $a=(u+v)/2$, $b=(u-v)/2$, the reverse sum-to-product identities follow by solving these formulas for the sums. The conventions and prerequisite facts used below are recorded in The addition formulas for sine and cosine, The subtraction formulas for sine and cosine.

If $t=\tan(\theta/2)$ is defined, then $\cos\theta=\frac{1-t^2}{1+t^2},\qquad \sin\theta=\frac{2t}{1+t^2}.$ Conversely every point $(x,y)$ on $x^2+y^2=1$ with $x\ne-1$ has the unique parameter $t=y/(1+x)$ and equals $((1-t^2)/(1+t^2),2t/(1+t^2))$. The conventions and prerequisite facts used below are recorded in Double-angle and quadratic power-reduction identities, Tangent, cotangent, secant, and cosecant on their exact natural domains, Parity and the Pythagorean identity for sine and cosine.

The map $t\mapsto(\cos t,\sin t)$ is a bijection from $[0,2\pi)$ onto $S^1=\{(x,y)\in\mathbb R^2:x^2+y^2=1\}$. The conventions and prerequisite facts used below are recorded in Parity and the Pythagorean identity for sine and cosine, Signs, monotonicity intervals, and ranges of sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi, Pi as twice the smallest positive zero of cosine.

For reals $a,b$ not both zero, put $R=\sqrt{a^2+b^2}>0$. There is a unique $\phi\in[0,2\pi)$ with $\cos\phi=a/R$ and $\sin\phi=b/R$, and $a\cos x+b\sin x=R\cos(x-\phi)$ for every real $x$. For $a=b=0$ the left side is identically zero. The conventions and prerequisite facts used below are recorded in $t\mapsto(\cos t,\sin t)$ is a bijection from $[0,2\pi)$ onto the real unit circle, Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$; the positives are $\{x^2 : x \neq 0\}$, The addition formulas for sine and cosine.

A formal real polynomial is either the zero polynomial $0$, or a finite coefficient list $(a_0,\ldots,a_n)$ with $a_n\ne0$; we write the latter as $p(X)=\sum_{k=0}^{n}a_kX^k$. The list, rather than the function it induces, is the polynomial object. Its evaluation at $x\in\mathbb R$ is the real number $p(x):=\sum_{k=0}^{n}a_kx^k$.

For nonzero $p=(a_0,\ldots,a_n)$, define $\deg p:=n$ and $\operatorname{lc}(p):=a_n$. The zero polynomial has no degree and no leading coefficient. A nonzero polynomial is monic when $\operatorname{lc}(p)=1$. Thus degree and leading coefficient are defined from the displayed finite list, without asserting that distinct formal polynomials define distinct functions. The conventions for finite sums and integer powers used in the evaluation are recorded in Finite sums and finite products, by recursion, Integer powers $a^m$.

If $p(x)=\sum_{k<n}a_kx^k$ and $p(a)=0$, then there is a real polynomial $q$ with $p(x)=(x-a)q(x)$ for every real $x$. The conventions and prerequisite facts used below are recorded in Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials, Factorisation of $b^n - a^n$, and the resulting Lipschitz estimate, Laws of finite sums and finite products.

A nonzero real polynomial of degree $n$ has at most $n$ distinct real roots. The conventions and prerequisite facts used below are recorded in Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials, A real polynomial vanishing at $a$ is divisible by $x-a$, The principle of mathematical induction.

Let $\mathcal P_{\mathbb R}$ be the set of formal real polynomials of Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials. Associate to every $P\in\mathcal P_{\mathbb R}$ an eventually zero coefficient sequence $(p_k)$: use its displayed finite coefficient list and extend it by zeros when $P\ne0$, and put $p_k=0$ for every $k$ when $P=0$. Do the same for $Q$, with coefficients $(q_k)$. Define $P+Q$ coefficientwise and define $PQ$ by the convolution coefficients

using the real finite sum of Finite sums and finite products, by recursion. In either construction, discard trailing zero coefficients; if every coefficient is zero, the result is the zero polynomial. Scalar multiplication and subtraction are the corresponding coefficientwise operations. Thus these formulas define operations on the formal coefficient-list objects, independently of evaluation.

Write $1=(1)$ and $X=(0,1)$, and define

Apply The recursion theorem to the set $\mathcal P_{\mathbb R}\times\mathcal P_{\mathbb R}$, first with initial value $(1,X)$ and then with initial value $(1,2X)$, always using the function $\Phi$. This gives unique pair sequences $H^T,H^U$. Define $T_n$ and $U_n$ to be the first coordinates of $H^T_n$ and $H^U_n$, respectively. Since the first coordinate of $\Phi(P,Q)$ is $Q$, the second coordinate of $H^T_n$ is $T_{n+1}$ and the second coordinate of $H^U_n$ is $U_{n+1}$. Consequently

for every $n\in\mathbb N$. These unique sequences are the Chebyshev polynomials of the first and second kinds.

For $n\ge1$, $T_n$ has degree $n$ and leading coefficient $2^{n-1}$, while $U_n$ has degree $n$ and leading coefficient $2^n$. The conventions and prerequisite facts used below are recorded in Chebyshev polynomials of the first and second kinds by their three-term recurrences, The principle of mathematical induction, Integer powers $a^m$.

For every $n\in\mathbb N$ and real $\theta$, $T_n(\cos\theta)=\cos(n\theta),\qquad U_n(\cos\theta)\sin\theta=\sin((n+1)\theta).$ In particular, for $n\ge1$ and $0\le j\le n$, $T_n(\cos(j\pi/n))=(-1)^j$. The conventions and prerequisite facts used below are recorded in Chebyshev polynomials of the first and second kinds by their three-term recurrences, The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, The principle of mathematical induction.

For every $n\in\mathbb N$ and real $\theta$, $\cos(n\theta)=\sum_{2j\le n}(-1)^j\binom n{2j}\cos^{n-2j}\theta\sin^{2j}\theta,$ $\sin(n\theta)=\sum_{2j+1\le n}(-1)^j\binom n{2j+1}\cos^{n-2j-1}\theta\sin^{2j+1}\theta.$ The conventions and prerequisite facts used below are recorded in The addition formulas for sine and cosine, Pascal's rule $\binom{n+1}{k+1} = \binom{n}{k} + \binom{n}{k+1}$, and the hockey-stick identity $\sum_{i \le n}\binom{i}{k} = \binom{n+1}{k+1}$, The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$, Finite sums and finite products, by recursion, Integer powers $a^m$, The principle of mathematical induction.

For $n\ge1$, the polynomial $P_n=2^{1-n}T_n$ is monic of degree $n$ and for every monic real polynomial $q$ of degree $n$, $\max_{x\in[-1,1]}|q(x)|\ge2^{1-n}=\max_{x\in[-1,1]}|P_n(x)|.$ Equality is attained by $P_n$. The conventions and prerequisite facts used below are recorded in Chebyshev polynomials of the first and second kinds by their three-term recurrences, Degrees and leading coefficients of the Chebyshev polynomials, $T_n(\cos\theta)=\cos(n\theta)$ and $U_n(\cos\theta)\sin\theta=\sin((n+1)\theta)$ for every $n\in\mathbb N$, Signs, monotonicity intervals, and ranges of sine and cosine, A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots, Formal real polynomials, evaluation, degree, leading coefficient, and monic polynomials, Extreme value theorem: a continuous real function on a nonempty compact subset of $\mathbb{R}$ attains a greatest and a least value, Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, and Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a,b]$ takes every value between $f(a)$ and $f(b)$.