Trigonometric Identities

I. Introduction: Trigonometric identities are equalities involving trigonometric functions that are true for all values of the variables within their domains. Understanding these identities is essential for simplifying expressions, solving equations, and proving mathematical relationships.

II. Pythagorean Identities:

  1. Pythagorean Identity:

    • sin2θ+cos2θ=1
    • This fundamental identity holds for all values of θ and is derived from the Pythagorean theorem in a right-angled triangle.
  2. Reciprocal Pythagorean Identities:

    • csc2θ=1+cot2θ
    • sec2θ=1+tan2θ
    • cot2θ=csc2θ1
    • tan2θ=sec2θ1

III. Reciprocal Identities:

  1. Cosecant (csc) Reciprocal Identity:

    • cscθ=1sinθ
  2. Secant (sec) Reciprocal Identity:

    • secθ=1cosθ
  3. Cotangent (cot) Reciprocal Identity:

    • cotθ=1tanθ

IV. Quotient and Co-Function Identities:

  1. Tangent (tan) Quotient Identity:

    • tanθ=sinθcosθ
  2. Cotangent (cot) Quotient Identity:

    • cotθ=cosθsinθ
  3. Secant (sec) and Cosecant (csc) Co-Function Identities:

    • secθ=1cosθ
    • cscθ=1sinθ

V. Double Angle Formulas:

  1. sin(2θ)=2sinθcosθ
  2. cos(2θ)=cos2θsin2θ
  3. tan(2θ)=2tanθ1tan2θ

VI. Half-Angle Formulas:

  1. sin(θ2)=±1cosθ2
  2. cos(θ2)=±1+cosθ2
  3. tan(θ2)=±1cosθ1+cosθ

VII. Sum and Difference Formulas:

  1. sin(A+B)=sinAcosB+cosAsinB
  2. cos(A+B)=cosAcosBsinAsinB
  3. tan(A+B)=tanA+tanB1tanAtanB