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Trigonometry In Mathematics, Exercises of Mathematics

Mathematics

Trigonometry In Mathematics ()

Typology: Exercises

2025/2026

Uploaded on 03/19/2026

koko-ali-firdaus-alfaridzi
koko-ali-firdaus-alfaridzi 🇮🇩

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Identities in Trigonometry
Koko Ali FA
March 15, 2026
1 Simple Trig Identities
1.1 Definition Identities
Theorem 1.1.1
•tan(x) = sin(x)
cos(x)
•sin(x) = 1
csc(x)
•cos(x) = 1
sec(x)
•tan(x) = 1
cot(x)
Theorem 1.1.2 (Pythagorean Identities)
•sin2(θ) + cos2(θ) = 1
•1 + tan2(θ) = sec2(θ)
•1 + cot2(θ) = csc2(θ)
2 More Trig Identities
2.1 Sum and Difference of Angles
Theorem 1.2.1 (Sum of Angles)
•sin(x+y) = sin(x) cos(y) + sin(y) cos(x)
•cos(x+y) = cos(x) cos(y)−sin(x) sin(y)
•tan(x+y) = tan(x)+tan(y)
1−tan(x) tan(y)
Theorem 1.2.2 (Difference of Angles)
•sin(x−y) = sin(x) cos(y)−sin(y) cos(x)
•cos(x−y) = cos(x) cos(y) + sin(x) sin(y)
•tan(x−y) = tan(x)−tan(y)
1+tan(x) tan(y)
1
pf3

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Identities in Trigonometry

Koko Ali FA

March 15, 2026

1 Simple Trig Identities

1.1 Definition Identities

Theorem 1.1.

  • tan(x) = (^) cos(sin(xx))
  • sin(x) = (^) csc(^1 x)
  • cos(x) = (^) sec(^1 x)
  • tan(x) = (^) cot(^1 x)

Theorem 1.1.2 (Pythagorean Identities)

  • sin^2 (θ) + cos^2 (θ) = 1
  • 1 + tan^2 (θ) = sec^2 (θ)
  • 1 + cot^2 (θ) = csc^2 (θ)

2 More Trig Identities

2.1 Sum and Difference of Angles

Theorem 1.2.1 (Sum of Angles)

  • sin(x + y) = sin(x) cos(y) + sin(y) cos(x)
  • cos(x + y) = cos(x) cos(y) − sin(x) sin(y)
  • tan(x + y) = 1 tan(−tan(x)+tan(x) tan(yy))

Theorem 1.2.2 (Difference of Angles)

  • sin(x − y) = sin(x) cos(y) − sin(y) cos(x)
  • cos(x − y) = cos(x) cos(y) + sin(x) sin(y)
  • tan(x − y) = (^) 1+tan(tan(x)x−) tan(tan(yy))

2.2 Double and Half Angle

Theorem 1.3.1 (Double Angle)

  • sin(2x) = 2 sin(x) cos(x)
  • cos(2x) = cos^2 (x) − sin^2 (x) = 2 cos^2 (x) − 1 = 1 − 2 sin^2 (x)
  • tan(2x) = (^1) −2 tan(tan 2 x()x)

Theorem 1.3.2 (Half Angle)

Half Angle:

  • sin( x 2 ) = ±

1 −cos(x) 2

  • cos( x 2 ) = ±

1+cos(x) 2

  • tan( x 2 ) = ±

1 −cos(x) 1+cos(x) =^

sin(x) 1+cos(x) =^

1 −cos(x) sin(x)

2.3 Sum to Product and Product to Sum

Theorem 1.4.1 (Sum to Product)

  • sin(x) + sin(y) = 2 sin( x+ 2 y) cos( x− 2 y)
  • sin(x) − sin(y) = 2 cos( x+ 2 y) sin( x− 2 y)
  • cos(x) + cos(y) = 2 cos( x+ 2 y) cos( x− 2 y)
  • cos(x) − cos(y) = −2 sin( x+ 2 y) sin( x− 2 y)

Theorem 1.4.2 (Product to Sum)

Product to Sum:

  • sin(x) sin(y) = − 12 (cos(x + y) − cos(x − y))
  • sin(x) cos(y) = 12 (sin(x + y) + sin(x − y))
  • cos(x) cos(y) = 12 (cos(x + y) + cos(x − y))