List of Trigonometric Identities

Basic Identities: Pythagorean and Ratios

• $sin² A + cos² A = 1$
• $tan² A + 1 = sec² A$
• $1 + cot² A = cosec² A$
• $tanA = {sinA} / {cosA}$
• $cosecA = 1 / {sinA}$
• $secA = 1 / {cosA}$
• $cotA = 1 / {tanA} = {cosA} / {sinA}$

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Co-function Identities: Shifts

• $sin({π/2} - {A}) = cosA$

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  $sin({90°} - A) = cosA$
• $cos({π/2} - A) = sinA$

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  $cos(90° - A) = sinA$
• $tan({π/2} - A) = cotA$

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  $tan(90° - A) = cotA$

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Odd-Even Identities: Reflection

• $sin(-A) = - sinA$
• $cos(-A) = cosA$
• $tan(-A) = - tanA$

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Addition Formulas: Angle Sum and Difference

• $sin(A + B) = sinA·cosB + cosA·sinB$
• $sin(A - B) = sinA·cosB - cosA·sinB$
• $cos(A + B) = cosA·cosB - sinA·sinB$
• $cos(A - B) = cosA·cosB + sinA·sinB$
• $tan(A + B) = {( tanA + tanB )} / {( 1 - tanA·tanB )}$
• $tan(A - B) = {( tanA - tanB )} / {( 1 + tanA·tanB )}$

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Double-Angle Formulas

• $sin2A = 2·sinA·cosA$
• $cos2A = cos² A - sin² A$

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  $cos2A = 1 - 2·sin² A$

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  $cos2A = 2·cos² A - 1$
• $tan2A = { 2·tanA } / { 1 - tan² A }$

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Half-Angle Formulas

• $sin²({A/2}) = {1 - cosA}/2$

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  $sin({A/2}) = √{{ 1 - cosA }/2}$
• $cos²({A/2}) = {1 + cosA }/2$

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  $cos({A/2}) = √{{ 1 + cosA }/2}$
• $tan({A/2}) = { 1 - cosA } / {sinA}$

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  $tan({A/2}) = {sinA} / { 1 + cosA }$

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Product Formulas

• $2·sinA·cosB = sin(A + B) + sin(A - B)$
• $2·cosA·cosB = cos(A + B) + cos(A - B)$
• $2·cosA·sinB = sin(A + B) - sin(A - B)$
• $2·sinA·sinB = cos(A - B) - cos(A + B)$

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Factoring Formulas

• $sinA + sinB = 2·cos({ {A - B}/2 })·sin({ {A + B}/2})$
• $sinA - sinB = 2·cos({ {A + B}/2 })·sin({ {A - B}/2 })$
• $cosA + cosB = 2·cos({ {A + B}/2 })·cos({ {A - B}/2 })$
• $cosA - cosB = - 2·sin({ {A + B}/2 })·sin({ {A - B}/2 })$