Trigonometric Identities

Group A

sin²θ + cos²θ = 1

sec²θ = 1 + tan²θ

cosec²θ = 1 + cot²θ

sin(A + B) = sinAcosB + cosAsinB

sin(A - B) = sinAcosB - cosAsinB

cos(A + B) = cosAcosB - sinAsinB

cos(A - B) = cosAcosB + sinAsinB

tan(A + B) = ^{tanA + tanB}⁄_{1 - tanAtanB}

tan(A - B) = ^{tanA - tanB}⁄_{1 + tanAtanB}

Group B

sin2θ = 2sinθcosθ

cos2θ = cos²θ - sin²θ = 1 - 2sin²θ = 2cos²θ - 1

tan2θ = ^2tanθ⁄_{1 - tan²θ}

sinφ = 2sin^φ⁄₂cos^φ⁄₂

cosφ = cos²^φ⁄₂ - sin²^φ⁄₂ = 1 - 2sin²^φ⁄₂ = 2cos²^φ⁄₂ - 1

tanφ = ^{2tan^φ⁄₂}⁄_{1 - 2tan²^φ⁄₂}

Group C

sinC + sinD = 2sin^{(C + D)}⁄₂cos^{(C - D)}⁄₂

sinC - sinD = 2cos^{(C + D)}⁄₂sin^{(C - D)}⁄₂

cosC + cosD = 2cos^{(C + D)}⁄₂cos^{(C - D)}⁄₂

cosD - cosC = 2sin^{(C + D)}⁄₂sin^{(C - D)}⁄₂

2sinAcosB = sin(A + B) + sin(A - B)

2cosAsinB = sin(A + B) - sin(A - B)

2cosAcosB = cos(A + B) + cos(A - B)

2sinAsinB = cos(A - B) - cos(A + B)

Where: A = √ a² + b² and α = tan^-1(^b⁄_a) (0° < α < 90°)

asinθ + bcosθ = Asin(θ + α)

asinθ - bcosθ = Asin(θ - α)

acosθ + bsinθ = Acos(θ - α)

acosθ - bsinθ = Acos(θ + α)

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