Trigonometric identities and proof

An identity and an equation make different claims

An identity gives equal values for every input in its stated common domain. An equation asks which inputs make two expressions equal. For example, sin 2x = 2 sin x cos x is an identity for all real angles. The equation sin 2x = sin x holds only at selected angles; the double-angle identity helps find them but does not make every x a solution.

A successful numerical substitution can check an individual value. A proof must explain why the equality holds throughout the domain, using identities and valid algebra.

A geometric identity generates others

On the unit circle, the point (cos x, sin x) is one unit from the origin. Pythagoras gives

Dividing every term by cos²x gives 1 + tan² x = sec² x, where cos x ≠ 0. Dividing instead by sin²x gives 1 + cot² x = cosec²x, where sin x ≠ 0. The original Pythagorean identity holds at all angles, but these divided forms keep their own denominator restrictions.

For example, at x = 0 the original says 0 + 1 = 1. The cosecant form is undefined there; writing it from the original identity has not made division by zero valid.

Angle addition keeps its signs

For any real angles A and B,

Sine keeps the sign between the angles. Cosine reverses it: cos(A − B) has a positive sin A sin B term. These products, rather than adding the separate sine or cosine values, account for the changed direction on the circle.

Dividing the sine expansion by the cosine expansion gives

This version requires tan A, tan B, tan(A ± B) and its displayed denominator all to be defined. For example, A = 30°, B = 60° makes the addition denominator zero: tan 90° is undefined, not a very large exact number.

Build an exact value from known angles

Use 75° = 45° + 30°:

Substituting the special-angle values gives (square root of 6 + square root of 2)/4. Both terms are positive because the sine-addition formula has a plus sign here.

For 15° = 45° − 30°, cosine subtraction gives

so its exact value is also (square root of 6 + square root of 2)/4. The equality agrees with sin 75° = cos 15°, since the angles are complementary. Tangent subtraction instead gives

Keep these exact forms when requested; rounded calculator values are checks rather than replacements for the derivation.

Choose a double-angle form that matches the expression

Putting B = A in the addition identities gives sin 2A = 2 sin A cos A and cos 2A = cos² A- sin² A. Using sin²A + cos²A = 1 produces two other useful forms:

For example, 1 − cos 2A becomes 2 sin²A, whereas 1 + cos 2A becomes 2 cos²A. Choose the version that creates the factors needed by the rest of the expression.

Also,

on the common domain. The right side needs cos A ≠ 0 as well as 1 − tan²A ≠ 0. At A = 90°, tan 2A = tan 180° = 0, but the right side is undefined because tan 90° is undefined. This formula cannot be used at that input even though the original tan 2A has a value there.

Transform one side without assuming the result

To prove (1- cos 2x)/sin 2x = tan x, begin by stating the original fraction’s condition sin 2x ≠ 0. Since sin 2x = 2 sin x cos x, neither sin x nor cos x can be zero there. Now transform the left side:

This is tan x, as required, and the cancellation was justified by the original restriction. At x = 0 the simplified tan x has value 0, but the original quotient is undefined. The proof establishes equality on the common domain; it does not fill the original expression’s holes.

Starting by assuming the proposed equality and reaching a true statement is insufficient if any step was not reversible. Transforming one side into the other with its conditions visible avoids that circular reasoning. A valid reversible transformation of both sides can also prove an identity, provided every step is justified.

A false identity can match isolated inputs

The claim sin(2x) = 2 sin x works at x = 0 but fails at x = π/2: its sides are 0 and 2. The correct formula includes cos x. One counterexample disproves a universal claim, whereas one successful substitution does not establish it.