Trigonometry: Identities and Equations

Trigonometric identity simplification

Trigonometric identity simplification means rewriting an expression in a shorter or known form, such as a constant, a single ratio, or one function of a single angle, by using identities that hold for every admissible angle. The goal is a form that can be evaluated at once. Three routes do most of the work: (1) reduce every function to sine and cosine, (2) use the Pythagorean family to convert squares, and (3) use angle-transformation formulas such as double-angle, sum-to-product and product-to-sum formulas.

The Pythagorean identities are \sin^2\theta+\cos^2\theta=1, 1+\tan^2\theta=\sec^2\theta and 1+\cot^2\theta=\operatorname{cosec}^2\theta. The last two follow by dividing the first by \cos^2\theta and by \sin^2\theta. Since \tan\theta=\dfrac{\sin\theta}{\cos\theta}, \cot\theta=\dfrac{\cos\theta}{\sin\theta}, \sec\theta=\dfrac{1}{\cos\theta} and \operatorname{cosec}\theta=\dfrac{1}{\sin\theta}, any mixed ratio can be written in sine and cosine. For example, \dfrac{\sec^2\theta}{\operatorname{cosec}^2\theta}=\tan^2\theta follows directly from these definitions.

A product of factors such as (1+\cot\theta-\operatorname{cosec}\theta)(1+\tan\theta+\sec\theta) is simplified by writing each factor over its own denominator, \sin\theta or \cos\theta. The numerators then combine into a difference of squares, (\sin\theta+\cos\theta)^2-1, and the identity (\sin\theta+\cos\theta)^2=1+2\sin\theta\cos\theta cancels the common denominator exactly, so the product is a constant.

A ratio such as \dfrac{1-\tan\theta}{1-\cot\theta} simplifies once 1-\cot\theta is written as \dfrac{\tan\theta-1}{\tan\theta}. The ratio then equals -\tan\theta, not \tan\theta, so the sign must be tracked carefully. Terms that reduce to the same function subtract to zero, so compare the simplified forms before expanding anything.

Double-angle forms in tangent are \cos2A=\dfrac{1-\tan^2A}{1+\tan^2A} and \sin2A=\dfrac{2\tan A}{1+\tan^2A}. If only \tan A is given, substitute its square directly and simplify the resulting fraction.

The same pattern gives \sec2\gamma=\dfrac{1+t^2}{1-t^2} with t=\tan\gamma. To match a given rational expression, cross-multiply, solve the resulting linear equation in t^2, and check that t^2\ge 0, since a negative value means no real angle exists.

Vieta's formulas link roots to coefficients. If \sin\theta and \cos\theta are the roots of ax^2+bx+c=0, then \sin\theta+\cos\theta=-\dfrac{b}{a} and \sin\theta\cos\theta=\dfrac{c}{a}. Squaring the sum and using (\sin\theta+\cos\theta)^2=1+2\sin\theta\cos\theta eliminates \theta and leaves a relation among a, b and c.

In any triangle A+B+C=\pi, which gives the identity \cos2A+\cos2B+\cos2C=-1-4\cos A\cos B\cos C. A condition such as \cos2A+\cos2B+\cos2C=-1 therefore forces \cos A\cos B\cos C=0. A quick check with an equilateral triangle confirms the identity, since both sides equal -\tfrac{3}{2}.

Since \tan^2\theta+\cot^2\theta=(\tan\theta+\cot\theta)^2-2, a sum of squares of tangent and cotangent reduces to a single quantity. For 15^\circ, \tan15^\circ=2-\sqrt3 and \cot15^\circ=2+\sqrt3, so the two squares are 7\mp4\sqrt3 and their sum is rational.

Fourth powers are reduced with a^4+b^4=(a^2+b^2)^2-2a^2b^2. Reduction formulas first turn \cos(\pi-x) into -\cos x and \cos(\tfrac{\pi}{2}+x) into -\sin x. Then \sin^2x+\cos^2x=1 gives \cos^4x+\sin^4x=1-\tfrac{1}{2}\sin^2 2x, so the value depends only on the double angle.

The difference of squares of sines has the product form \sin^2A-\sin^2B=\sin(A+B)\sin(A-B), which is often faster than expanding both squares. The same result follows from \sin^2x=\tfrac{1}{2}(1-\cos2x).

Product-to-sum formulas, 2\cos X\cos Y=\cos(X+Y)+\cos(X-Y) and 2\sin X\cos Y=\sin(X+Y)+\sin(X-Y), convert products into sums. When the resulting angles add to \pi, the relation \cos(\pi-x)=-\cos x makes terms cancel in pairs.

Periodicity and parity come first: \sin(\theta+2n\pi)=\sin\theta for every integer n, sine is odd so \sin(-\theta)=-\sin\theta, and cosine is even. Applying these rules before multiplying prevents sign mistakes. For instance, \sin(\pi-\theta) and \tan(\pi-\theta) differ in sign and must be kept separate.

Since \tan(180^\circ-\theta)=-\tan\theta and \cot(180^\circ-\theta)=-\cot\theta, the squares of tangent and cotangent are unchanged by this reflection. For example, \tan^2150^\circ=\tan^230^\circ=\dfrac{1}{3}.

The triple-angle identities are \sin3\theta=3\sin\theta-4\sin^3\theta and \cos3\theta=4\cos^3\theta-3\cos\theta. A polynomial in \sin\theta or \cos\theta that matches one of these brackets, possibly up to a sign, reduces to a single triple-angle function.

Substitution t=\sin x+\cos x is effective when an expression contains sums and products of sine and cosine. Since t^2=1+\sin2x and \sin x\cos x=\tfrac{t^2-1}{2}, the equation converts into a polynomial in t, and the result is then expressed in \sin2x.

If x+\dfrac{1}{x}=2\cos\theta, then x^n+\dfrac{1}{x^n}=2\cos n\theta for integer n. Cubing gives x^3+\dfrac{1}{x^3}=\left(x+\dfrac{1}{x}\right)^3-3\left(x+\dfrac{1}{x}\right), and substituting the cosine form together with the triple-angle identity completes the reduction.

The sum-to-product formulas are \sin A+\sin B=2\sin\dfrac{A+B}{2}\cos\dfrac{A-B}{2} and \cos A+\cos B=2\cos\dfrac{A+B}{2}\cos\dfrac{A-B}{2}. Dividing one by the other cancels the common factor and gives \tan\dfrac{A+B}{2}, so a ratio of sum forms is a tangent of the half-sum, not of the half-difference.

In a triangle with C=60^\circ, the relation A+B=120^\circ holds, so \cos A+\cos B reduces to \cos\dfrac{A-B}{2} once \cos60^\circ=\tfrac{1}{2} is used. The common factor then cancels, and the ratio equals 1 because the factor is nonzero in a triangle.

A sum of products of sines at several angles is handled by first using \sin(180^\circ+x)=-\sin x to bring the angles into a standard range. Product-to-sum or sum-to-product formulas are then applied to pairs whose angles combine neatly, and \sin^2x=\tfrac{1}{2}(1-\cos2x) finishes the reduction.

For p=\sin A+\sin B and q=\cos A+\cos B, one finds p^2+q^2=2+2\cos(A-B), while p^2-q^2 contains the factor \cos(A+B). The quotient \dfrac{p^2-q^2}{p^2+q^2} therefore equals -\cos(A+B), which is \cos(\pi-A-B). The relation \cos\left(\tfrac{\pi}{2}-x\right)=\sin x is different and should not be confused with it.

An expression a\cos x\pm b\sin x can be rewritten as a single cosine or sine with amplitude \sqrt{a^2+b^2}. For example, \sqrt3\cos\theta-\sin\theta=2\sin(60^\circ-\theta), because \sin60^\circ=\tfrac{\sqrt3}{2} and \cos60^\circ=\tfrac{1}{2}. Pairing this with \sin2x=2\sin x\cos x in a denominator reduces the ratio to a simple quotient of sines.

Squaring a difference such as \sin\theta-\cos\theta gives 1-\sin2\theta, because (\sin\theta-\cos\theta)^2=1-2\sin\theta\cos\theta. Taking the square root then requires the sign of the original difference, which depends on the quadrant or on which function is larger.

The identity \dfrac{1}{\sin^2x}+\dfrac{1}{\cos^2x}=\dfrac{1}{\sin^2x\cos^2x} follows by adding the fractions, because \sin^2x+\cos^2x=1. Recognising this pattern lets a ratio of two equal expressions be simplified to 1 without further work.

To summarise, a simplification should start by identifying the family of identities involved, then convert to a common basis such as sine and cosine, and finally check the sign and the domain of any square root or tangent that appears.

Trigonometric equations

A trigonometric equation is an equation in which the unknown appears inside a trigonometric function, and its solutions are all the angles that satisfy it. Because sine, cosine and tangent are periodic, the complete answer is a general solution containing an integer n, and an interval such as 0<x<2\pi selects the values that lie inside it. Before solving, check that the right-hand side lies in the range of the function: \sin x=k is possible only when -1\le k\le 1, which turns a question about a parameter into an inequality. The basic equations are \sin A=\sin B, giving A=n\pi+(-1)^nB, and \cos A=\cos B, giving A=2n\pi\pm B. A co-function identity such as \cos 3\theta=\sin(90^\circ-3\theta) converts a mixed equation into one function of a single angle, after which the basic equation applies. An equation that is quadratic in \sin x or \cos x is solved by substitution: solve the quadratic in the new variable and reject every root outside [-1,1]. Within a given interval, the roots for \sin^2x=\frac34 are \frac{\pi}{3} and \frac{2\pi}{3}, and this method applies equally to quadratics in a multiple angle such as \sin 2x, where only the root with modulus at most one is admissible. When the interval is stated for a multiple angle, count the solutions after the substitution, because the new variable covers a longer range. For \cos 2x=\frac12 with 0<x<360^\circ, the angle 2x runs over a range twice as long, which produces four values of x. The same care is needed for \sin 2\theta=-1 on (-\pi,\pi), which has exactly two solutions in that interval. Multiplying both sides by a function can add or remove solutions. Multiplying by \cos x is valid only where \cos x\neq 0, so the excluded point must be checked separately; the result is then a quadratic in \sin x, and roots lying outside the interval are rejected. A logarithmic equation \log_b a=c means a=b^c, with b>0, b\neq 1 and a>0. For c=1 this reduces to a=b, so the base conditions must be verified after the solution is found. Sums of sines or cosines are factorised with the sum-to-product formulas \sin P+\sin Q=2\sin\frac{P+Q}{2}\cos\frac{P-Q}{2} and \cos P+\cos Q=2\cos\frac{P+Q}{2}\cos\frac{P-Q}{2}. A common factor may be cancelled only when it is non-zero, and the remaining equation gives the tangent of the half-sum of the angles. An equation of the form \tan A=\cot B is solved by writing \cot B=\tan\left(\frac{\pi}{2}-B\right), which gives A=n\pi+\frac{\pi}{2}-B for an integer n. Combining this with the angle sum leads to a linear relation \sin\theta+\cos\theta=n+\frac12, and since 8\sin^2\left(\theta+\frac{\pi}{4}\right)=4(1+\sin 2\theta)=4(\sin\theta+\cos\theta)^2, the requested value follows from the admissible n. When two angles are determined only up to sign choices, a linear combination of them takes at most as many values as there are distinct sign combinations. Each combination must be computed and the results compared, because different combinations can coincide.

Tangent addition formula

The tangent addition formula states \tan(A+B)=\dfrac{\tan A+\tan B}{1-\tan A\tan B} and \tan(A-B)=\dfrac{\tan A-\tan B}{1+\tan A\tan B}, valid whenever the tangents are defined and the denominator is non-zero. Since \tan 45^\circ=1, the special case \tan(45^\circ+\theta)=\dfrac{1+\tan\theta}{1-\tan\theta} is used frequently. When \tan\theta is known, this is evaluated by direct substitution, so that \tan\theta=\frac12 gives \frac{3/2}{1/2}=3. Writing the same ratio as \dfrac{\cos\theta+\sin\theta}{\cos\theta-\sin\theta} and multiplying numerator and denominator by \cos\theta+\sin\theta gives \dfrac{1+\sin 2\theta}{\cos 2\theta}, because (\cos\theta+\sin\theta)^2=1+\sin2\theta and \cos^2\theta-\sin^2\theta=\cos2\theta. This form is useful when an equation links \tan(45^\circ+\theta) with a function of 2\theta. A relation of the form \tan A+\tan B=1-\tan A\tan B is recognised by reading the addition formula backwards. The numerator then equals the denominator, so \tan(A+B)=1, and A+B=n\pi+\frac{\pi}{4} for an integer n, provided \tan A\tan B\neq 1. Expanding a product such as (1+\tan\theta)(1+\tan9\theta) gives 1+\tan\theta+\tan9\theta+\tan\theta\tan9\theta, and if this equals a given constant, the relation between the two tangents has the form required for the addition formula. For A=\theta and B=9\theta, the result is \tan10\theta=1. An expression of the form \dfrac{f(\alpha)+f(\beta)}{1-f(\alpha)f(\beta)} is the tangent addition formula in disguise. For f(\alpha)=\sqrt{\sec^2\alpha-1}=|\tan\alpha|, the expression equals \tan(\alpha+\beta) when both tangents are non-negative, so it corresponds to f(\alpha+\beta). A square root of a trigonometric expression should therefore be reduced to an absolute value before the formula is identified. If \tan\alpha and \tan\beta are the roots of a quadratic x^2-px+q=0, then \tan\alpha+\tan\beta=p and \tan\alpha\tan\beta=q, so \tan(\alpha+\beta)=\dfrac{p}{1-q}. Combined with \cos2\theta=\dfrac{1-\tan^2\theta}{1+\tan^2\theta}, this gives the cosine of twice the compound angle without finding either angle separately.

Trigonometric ratios in quadrants

The sign of each trigonometric ratio depends on the quadrant in which the angle lies, and the rule is remembered as ASTC: all ratios are positive in the first quadrant, sine alone in the second, tangent alone in the third, and cosine alone in the fourth. Since \tan\theta=\dfrac{\sin\theta}{\cos\theta}, a negative tangent places the angle in the second or the fourth quadrant. Both sign possibilities for sine must then be considered, with the magnitude of sine found from a right triangle whose legs are the numerator and denominator of the tangent. The product of sine and cosine can be found from the tangent alone. Since \sin x\cos x=\tan x\cos^2x and \cos^2x=\dfrac{1}{1+\tan^2x}, we get \sin x\cos x=\dfrac{\tan x}{1+\tan^2x}, and the sign of the product is then fixed by the quadrant, which is negative in the second quadrant. Trigonometric ratios are periodic, so a large or negative angle is reduced by a multiple of the period before the quadrant rule is applied. Cosecant has period 2\pi, and because sine is an odd function, \operatorname{cosec}(-\theta)=-\operatorname{cosec}\theta. A negative angle is therefore brought into a standard range, after which the sign and the standard value are read off. Two ratios given together fix the position of the angle more precisely than either alone. Since \cot\theta=\dfrac{\cos\theta}{\sin\theta} and \operatorname{cosec}\theta=\dfrac{1}{\sin\theta}, a pair of conditions with given signs determines both \sin\theta and \cos\theta, and only one angle in a single period can satisfy both. In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. When one side is fixed and the other varies over admissible values, the tangent takes a corresponding set of values, and the admissible value is the one that matches the given geometry.