Trigonometric Identities

01 / UNDERSTAND & APPLY

1. An identity is an equality on its common domain

A trigonometric identity holds for every input where both displayed sides are defined. An equation to be solved holds only for selected inputs. Proving an identity and solving an equation therefore have different goals, even when they use the same algebraic transformations.

The unit circle gives sin²x+cos²x=1. Dividing by cos²x gives 1+tan²x=sec²x where cos x≠0. Dividing by sin²x gives 1+cot²x=csc²x where sin x≠0. The derived identities inherit these denominator restrictions.

Reciprocal and quotient definitions are tan x=sin x/cos x, cot x=cos x/sin x, sec x=1/cos x and csc x=1/sin x on their respective domains. Rewriting in sine and cosine often reveals a common denominator, but it does not make excluded inputs valid.

Unless a degree symbol is shown explicitly, angles here are in radians. A full sine or cosine period is 2π, and a tangent or cotangent period is π. Mixing degrees and radians in one calculation changes the actual angle rather than merely its notation.

02 / UNDERSTAND & APPLY

2. Reduce angles using geometry, not sign mnemonics alone

Sine is odd and cosine is even: sin(−x)=−sin x and cos(−x)=cos x. Tangent is odd on its domain. Unit-circle reflection gives these signs directly, and a half-turn gives sin(x+π)=−sin x and cos(x+π)=−cos x.

Cofunction identities include sin(π/2−x)=cos x and cos(π/2−x)=sin x. Combining them with periodicity and symmetry reduces angles to familiar intervals while preserving signs. For example, sin(π+x)=−sin x and cos(π−x)=−cos x.

A period T>0 satisfies f(x+T)=f(x) with the domain preserved. A common period of two functions is a period of their sum or product, but it need not be the least positive period after simplification. The expression sin²x+cos²x is constant and has no least positive period.

For a nonconstant sine or cosine with argument kx+b, k≠0, the fundamental period is 2π/|k|. Squared sine or cosine has period π/|k|. Tangent has period π/|k|, with its excluded poles shifting accordingly. For an equation, use a period of the entire equation rather than one term alone.

03 / UNDERSTAND & APPLY

3. Addition formulas describe rotations

A rotation through an angle u sends a unit vector at angle v to one at angle u+v. Multiplying the rotation matrix [[cos u,−sin u],[sin u,cos u]] by (cos v,sin v) gives cos(u+v)=cos u cos v−sin u sin v and sin(u+v)=sin u cos v+cos u sin v.

Replacing v by −v and using parity produces the subtraction formulas. The sine formula keeps the sign of the angle operation, while cosine changes it in the sine-product term. Deriving the pair from rotation is safer than memorizing an isolated sign pattern.

The tangent addition formula is tan(u+v)=(tan u+tan v)/(1−tan u tan v) when all displayed tangents and the denominator are defined. It follows by dividing the sine and cosine addition formulas. Cases with a zero denominator may describe a vertical direction, not an ordinary finite tangent value.

Addition formulas can evaluate exact nonstandard angles, combine oscillations or expose factors in equations. They do not imply sin(u+v)=sin u+sin v. The products involving both angles are essential.

04 / UNDERSTAND & APPLY

4. Match the identity to the desired variable

Setting the two angles equal gives sin2x=2sin x cos x and cos2x=cos²x−sin²x=1−2sin²x=2cos²x−1. The three cosine forms are interchangeable, but one may reduce an equation to a single trigonometric function more efficiently.

Power reduction gives sin²x=(1−cos2x)/2 and cos²x=(1+cos2x)/2. Half-angle square roots therefore involve signs: sin(x/2)=±√[(1−cos x)/2] and cos(x/2)=±√[(1+cos x)/2], with the sign determined by the quadrant of x/2.

The identity tan(x/2)=sin x/(1+cos x) holds where 1+cos x≠0, exactly the excluded odd multiples of π for the half-angle tangent. The alternative (1−cos x)/sin x needs sin x≠0 and thus has additional exclusions where the first form may still work.

Higher powers often simplify through the Pythagorean identity before any angle expansion. For instance, sin⁴x+cos⁴x=(sin²x+cos²x)²−2sin²x cos²x=1−(1/2)sin²2x, immediately showing its range [1/2,1].

05 / UNDERSTAND & APPLY

5. Convert sums to factors or to one shifted wave

Adding the sine addition and subtraction formulas gives sin u+sin v=2sin((u+v)/2)cos((u−v)/2). Subtracting gives sin u−sin v=2cos((u+v)/2)sin((u−v)/2). Analogous formulas convert cosine sums and differences into products.

Products can also be converted to sums: 2sin u cos v=sin(u+v)+sin(u−v), 2cos u cos v=cos(u+v)+cos(u−v), and 2sin u sin v=cos(u−v)−cos(u+v). These forms are especially useful when frequencies differ.

For real A,B not both zero, set R=√(A²+B²). Choose φ with cos φ=A/R and sin φ=B/R. Then A sin x+B cos x=R sin(x+φ), so its exact range is [−R,R]. Using only tan φ=B/A without quadrant information can choose the wrong phase.

The amplitude form turns a linear combination into one standard equation and provides an immediate feasibility test. If A=B=0, the expression is identically zero and must be treated separately; dividing by R would then be invalid.

06 / UNDERSTAND & APPLY

6. Prove a common-domain identity without assuming the result

A reliable identity proof starts from one side and transforms it into the other using equalities valid on the stated domain. Alternatively transform both sides independently into the same expression. Starting by assuming the desired equality and deriving a true statement is not automatically a proof unless every step is reversible.

Common strategies are converting to sine and cosine, factoring with 1−sin²x or 1−cos²x, multiplying by a conjugate and finding a common denominator. Before canceling a factor, note where it vanishes. A simplified expression may extend beyond the original domain, but the original identity does not automatically gain those inputs.

One numerical counterexample disproves a claimed identity. Many matching numerical values do not prove it. Symbolic derivations or a complete structural argument are needed to establish the equality for all allowed x.

07 / UNDERSTAND & APPLY

7. General solutions require every periodic family

For |c|≤1, let α=arcsin c in [−π/2,π/2]. Then sin x=c has solutions x=α+2kπ or x=π−α+2kπ, k∈ℤ. At c=±1 the two families coincide, so do not double-count them.

For |c|≤1, let β=arccos c in [0,π]. Then cos x=c has solutions x=±β+2kπ. For any real c, tan x=c gives x=arctan c+kπ. The tangent domain excludes its poles automatically, but any additional original denominator restrictions still apply.

For a multiple-angle equation, solve for the entire angle first, including its period, then divide. For sin(3x)=1/2, there are two angle families and division by 3 also divides their period. Solving only 3x=π/6 would omit infinitely many solutions.

If an interval is specified, enumerate the general families within it, check whether endpoints are included and remove duplicates. A half-open interval such as [0,2π) avoids counting the same periodic endpoint twice. A closed interval includes both endpoints if both satisfy the equation.

08 / UNDERSTAND & APPLY

8. Factor before dividing by a trigonometric expression

An equation such as sin2x=sin x becomes sin x(2cos x−1)=0. Solve each factor separately. Dividing by sin x would lose every solution with sin x=0, which is a whole legitimate family.

If an equation is polynomial in sin x or cos x, set u equal to that function and solve the polynomial, then retain only u∈[−1,1]. A polynomial root outside that interval does not correspond to a real angle. For a polynomial in tan x, real u values are allowed but original tangent poles remain excluded.

Identities can bring an equation to one variable. For example, replace cos2x with 1−2sin²x when the other terms involve sin x. Expanding every angle indiscriminately can create more variables and hide a simple factorization.

When an equation has a product equal to zero, each factor must be defined before the zero-product rule is used. An undefined tangent factor multiplied by a zero sine is still undefined, not automatically zero.

09 / UNDERSTAND & APPLY

9. Necessary transformations need a final original check

Squaring preserves equality but can lose sign information. If sin x+cos x=1, squaring gives 1+sin2x=1, hence sin2x=0. The squared equation also accepts inputs where the original sum is −1. Recheck every candidate in the original equation or retain the sign condition throughout.

For rational expressions in sine and cosine, t=tan(x/2) gives sin x=2t/(1+t²) and cos x=(1−t²)/(1+t²). This converts many equations to rational algebra, but excludes odd multiples of π where t is undefined. Check those inputs separately in the original equation.

After solving for t, recover x=2arctan t+2kπ, not merely x=2arctan t if all real solutions are requested. Any denominators introduced or cleared during the rational calculation must also be checked.

For equations involving tan and cot, a sine-cosine rewrite often exposes their common domain and makes the algebra shorter. Multiplying by sin x cos x is reversible only after confirming it is nonzero on that original domain.

10 / UNDERSTAND & APPLY

10. Solve on one period, then repeat with the correct boundaries

Trigonometric inequalities are interval problems. Locate equality points and domain exclusions on one full period, determine the sign between consecutive boundaries and then repeat. Isolated sample values identify a sign only after continuity on that interval is established.

For sin x≥1/2 on [0,2π), the solution is [π/6,5π/6]. For tan x>1, the solution is (π/4+kπ,π/2+kπ), k∈ℤ. The tangent pole is always excluded, even if an algebraic manipulation suggests a closed endpoint.

An amplitude equation A sin x+B cos x=c with R=√(A²+B²)>0 has no real solutions if |c|>R. In one full half-open period of the resulting nonconstant wave, it has one solution if |c|=R and two if |c|<R. If a nonunit frequency is present, count its periods within the requested interval.

The case A=B=0 is separate: 0=c gives every real x when c=0 and none otherwise. Parameter values that collapse a trigonometric expression to a constant can change both its period and its root count.

11 / UNDERSTAND & APPLY

11. Original practice: identities and exact values

Give the domain whenever a denominator or tangent appears. Determine half-angle signs from the actual angle interval.

12 / UNDERSTAND & APPLY

12. Original practice: complete solution sets

All unrestricted solution families use k∈ℤ. For interval answers, respect the exact inclusion of endpoints.

13 / UNDERSTAND & APPLY

13. Every transformation has a job and a domain

Use Pythagorean identities to reduce powers, addition formulas to combine angles, product-sum formulas to expose factors and amplitude forms to isolate one wave. An identity is useful when it simplifies the structure of the particular problem, not merely because it is available.

For equations, keep all periodic families and check the requested interval. Factor before dividing, reject polynomial sine/cosine values outside [−1,1], and test candidates after squaring. A half-angle substitution has an omitted angle class that must be checked separately.

For inequalities, split at equality points and poles, and preserve strict versus non-strict endpoints. For parameter problems, include the values that make an amplitude zero or a frequency vanish instead of applying a nonconstant formula blindly.

The final audit asks three questions: is every reported input in the original domain, does it satisfy the original equation, and has every possible periodic or factored branch been considered? Those checks make a short trigonometric answer complete.