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In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles .
Identity 1: sin 2 θ + cos 2 θ = 1 {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1} The following two results follow from this and the ratio identities.
Trigonometric identities may help simplify the answer. [ 1 ] [ 2 ] Like other methods of integration by substitution, when evaluating a definite integral, it may be simpler to completely deduce the antiderivative before applying the boundaries of integration.
Alternatively, the identities found at Trigonometric symmetry, shifts, and periodicity may be employed. By the periodicity identities we can say if the formula is true for −π < θ ≤ π then it is true for all real θ. Next we prove the identity in the range π/2 < θ ≤ π, to do this we let t = θ − π/2, t will now be in the range 0 ...
Substituting r(cos θ + i sin θ) for e ix and equating real and imaginary parts in this formula gives dr / dx = 0 and dθ / dx = 1. Thus, r is a constant, and θ is x + C for some constant C. The initial values r(0) = 1 and θ(0) = 0 come from e 0i = 1, giving r = 1 and θ = x.
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See angle sum and difference identities. We deduce that S(k) implies S(k + 1). By the principle of mathematical induction it follows that the result is true for all natural numbers. Now, S(0) is clearly true since cos(0x) + i sin(0x) = 1 + 0i = 1. Finally, for the negative integer cases, we consider an exponent of −n for natural n.