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  2. Proofs of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/Proofs_of_trigonometric...

    For example, the sine of angle θ is defined as being the length of the opposite side divided by the length of the hypotenuse. The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six ...

  3. List of trigonometric identities - Wikipedia

    en.wikipedia.org/wiki/List_of_trigonometric...

    These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function , and then simplifying the resulting integral with a trigonometric identity.

  4. Identity (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Identity_(mathematics)

    Visual proof of the Pythagorean identity: for any angle , the point (,) = (⁡, ⁡) lies on the unit circle, which satisfies the equation + =.Thus, ⁡ + ⁡ =. In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables ...

  5. De Moivre's formula - Wikipedia

    en.wikipedia.org/wiki/De_Moivre's_formula

    De Moivre's formula is a precursor to Euler's formula = ⁡ + ⁡, with x expressed in radians rather than degrees, which establishes the fundamental relationship between the trigonometric functions and the complex exponential function.

  6. Hockey-stick identity - Wikipedia

    en.wikipedia.org/wiki/Hockey-stick_identity

    The hockey stick identity confirms, for example: for n=6, r=2: 1+3+6+10+15=35. In combinatorics , the hockey-stick identity , [ 1 ] Christmas stocking identity , [ 2 ] boomerang identity , Fermat's identity or Chu's Theorem , [ 3 ] states that if n ≥ r ≥ 0 {\displaystyle n\geq r\geq 0} are integers, then

  7. Proofs That Really Count - Wikipedia

    en.wikipedia.org/wiki/Proofs_That_Really_Count

    Proofs That Really Count: the Art of Combinatorial Proof is an undergraduate-level mathematics book on combinatorial proofs of mathematical identies.That is, it concerns equations between two integer-valued formulas, shown to be equal either by showing that both sides of the equation count the same type of mathematical objects, or by finding a one-to-one correspondence between the different ...

  8. List of mathematical proofs - Wikipedia

    en.wikipedia.org/wiki/List_of_mathematical_proofs

    Fermat's little theorem and some proofs; Gödel's completeness theorem and its original proof; Mathematical induction and a proof; Proof that 0.999... equals 1; Proof that 22/7 exceeds π; Proof that e is irrational; Proof that π is irrational; Proof that the sum of the reciprocals of the primes diverges

  9. Sophomore's dream - Wikipedia

    en.wikipedia.org/wiki/Sophomore's_dream

    The proofs of the two identities are completely analogous, so only the proof of the second is presented here. The key ingredients of the proof are: to write x x = exp ⁡ ( x ln ⁡ x ) {\textstyle x^{x}=\exp(x\ln x)} (using the notation ln for the natural logarithm and exp for the exponential function );