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#### List of Trigonometric Identities

### Knowledge

Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined.

These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions.

##### Trigonometry Function Identities

###### Co-function Identities

sinθ = cos(π/2 - θ)

secθ = csc(π/2 - θ)

tanθ = cot(π/2 - θ)

###### Negative Angle Identities

sin(-θ) = - sinθ

cos(-θ) = cosθ

tan(-θ) = - tanθ

csc(-θ) = - cscθ

sec(-θ) = secθ

cot(-θ) = -cotθ

###### Addition and Subtraction Identities

sin( A + B ) = sinA cosB + cosA sinB

cos( A + B ) = cosA cosB - sinA sinB

tan( A + B ) = = *tanA + tanB**1 - tanA tanB*

sin( A - B ) = sinA cosB - cosA sinB

cos( A - B ) = cosA cosB - sinA sinB

tan( A - B ) = = *tanA - tanB**1 + tanA tanB*

###### Double-Angle Identities

sin2θ = 2 sinθ cosθ

cos2θ = cos^{2}θ - sin^{2}θ

cos2θ = 2cos^{2}θ - 1

cos2θ = 1 - 2sin^{2}θ

tan2θ = *2tanθ**1-tan ^{2}θ*

###### Product Identities

sinA cosB = *1**2*( sin(A+B) + sin(A-B) )

cosA sinB = *1**2*( sin(A+B) - sin(A-B) )

cosA cosB = *1**2*( cos(A+B) + cos(A-B) )

sinA sinB = *1**2*( cos(A-B) - cos(A+B) )

###### Supplement Angle Identities

sin( π - θ ) = sinθ

cos( π - θ ) = - cosθ

tan( π - θ ) = - tanθ

sin( π + θ ) = - sinθ

cos( π + θ ) = - cosθ

tan( π + θ ) = tanθ

csc( π - θ ) = cscθ

sec( π - θ ) = - secθ

cot( π - θ ) = - cotθ

csc( π + θ ) = - cscθ

sec( π + θ ) = - secθ

cot( π + θ ) = cotθ

###### Quotient Identities

tanθ = *sinθ**cosθ*

secθ = *1**cosθ*

cotθ = *cosθ**sinθ* = *1**tanθ*

cscθ = *1**sinθ*

###### Pythagorean Identities

sin^{2}θ + cos^{2}θ = 1

tan^{2}θ + 1 = sec^{2}θ

cot^{2}θ + 1 = csc^{2}θ

###### Half-Angle Identities

sin*θ**2* = ± *1-cosθ**2*

cos*θ**2* = ± *1+cosθ**2*

tan*θ**2* = ± *1-cosθ**1+cosθ*

###### Sum Identities

sinA + sinB = 2sin(*A+B**2*)cos(*A-B**2*)

sinA - sinB = 2cos(*A+B**2*)sin(*A-B**2*)

cosA + cosB = 2cos(*A+B**2*)cos(*A-B**2*)

cosA - cosB = -2cos(*A+B**2*)sin(*A-B**2*)

#### Example

θ_{1} is located in Quadrant III, and cos(θ_{1}) = - *3**5*

sin^{2}(θ_{1}) + cos^{2}(θ_{1}) = 1

sin^{2}(θ_{1}) + (- *3**5* )^{2} = 1

sin^{2}(θ_{1}) = 1 - (- *3**5* )^{2}

sin(θ_{1}) = ± *4**5*

The sign of sin(θ_{1}) is determined by the quadrant. θ_{1} is in Quadrant Ⅲ, so its sine value must be negative. In conclusion, sin(θ_{1}) = - *4**5* .