Short note
for
trigonometric
identies
1
mathematic
March 10, 2026
s
, 1. The Pythagorean Identities
These are derived from the equation of the unit circle (𝑥2 + 𝑦2 = 1). They are
arguably the most important identities in the subject.
• sin2(𝜃) + cos2(𝜃) = 1
• 1 + tan2(𝜃) = sec2(𝜃)
• 1 + cot2(𝜃) = csc2(𝜃)
2. Reciprocal and Quotient Identities
These define the relationships between the primary functions (sin, cos, tan)
and their reciprocals.
• Reciprocal:
o csc(𝜃) = o
sec(𝜃) = o
cot(𝜃) =
• Quotient:
o tan(𝜃) = o
cot(𝜃) =
3. Double-Angle Identities
These are vital for reducing the "power" of an expression or changing the
frequency of the input.
2
March 10, 2026
for
trigonometric
identies
1
mathematic
March 10, 2026
s
, 1. The Pythagorean Identities
These are derived from the equation of the unit circle (𝑥2 + 𝑦2 = 1). They are
arguably the most important identities in the subject.
• sin2(𝜃) + cos2(𝜃) = 1
• 1 + tan2(𝜃) = sec2(𝜃)
• 1 + cot2(𝜃) = csc2(𝜃)
2. Reciprocal and Quotient Identities
These define the relationships between the primary functions (sin, cos, tan)
and their reciprocals.
• Reciprocal:
o csc(𝜃) = o
sec(𝜃) = o
cot(𝜃) =
• Quotient:
o tan(𝜃) = o
cot(𝜃) =
3. Double-Angle Identities
These are vital for reducing the "power" of an expression or changing the
frequency of the input.
2
March 10, 2026
