Trigonometric identities

• Understanding trigonometric identities is crucial when tackling the Core Pure Mathematics 2 content. They allow you to simplify complex expressions, solve trigonometric equations, and prove mathematical relations.

• A fundamental set of identities to understand are the reciprocal identities. Cosecant (csc), secant (sec), and cotangent (cot) are the reciprocals of sine, cosine, and tangent respectively:
  1. csc x = 1 / sin x
  2. sec x = 1 / cos x
  3. cot x = 1 / tan x
• Alongside these, there are the quotient identities; these are especially important for dividing one trigonometric function by another:
  1. tan x = sin x / cos x
  2. cot x = cos x / sin x
• The Pythagorean identities are derived from the Pythagorean theorem and show relationships between sine, cosine, and tangent:
  1. sin²x + cos²x = 1
  2. 1 + tan²x = sec²x
  3. 1 + cot²x = csc²x
• Co-Function identities demonstrate the relationship between sine and cosine, and tangent and cotangent at complementary angles:
  1. sin(90° - x) = cos x
  2. cos(90° - x) = sin x
  3. tan(90° - x) = cot x
  4. cot(90° - x) = tan x
• Double angle identities express trigonometric functions of twice a certain angle:
  1. sin(2x) = 2sin x cos x
  2. cos(2x) = cos²x - sin²x = 2cos²x - 1 = 1 - 2sin²x
  3. tan(2x) = (2 tan x) / (1 - tan²x)
• Half angle identities express the trigonometric functions of half an angle x in terms of angles 2x:
  1. sin(x/2) = ±√[(1 - cos x) / 2]
  2. cos(x/2) = ±√[(1 + cos x) / 2]
  3. tan(x/2) = ±√[(1 - cos x) / (1 + cos x)]
• The sum and difference identities allow the expression of sin, cos, and tan of sum or difference of two angles in terms of simple functions:
  1. sin(A ± B) = sin A cos B ± cos A sin B
  2. cos(A ± B) = cos A cos B ∓ sin A sin B
  3. tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
• The understanding of these identities can be further deepened by solving related complex problems, and by learning how they can be applied in proofs and derivation of other identities.