Sum of the trigonometric functions: sinα + sinβ = 2sin((α+β)/2)cos((α-β)/2), cosα + cosβ = 2cos((α+β)/2)cos((α-β)/2), sinα + cosα = (square root of 2) sin(π/4 + α) = (square root of 2) cos(π/4 -α), 1 + cosα = 2cos^2(α/2), 1 + sinα = 2sin^2(π/4 + α/2) = 2cos^2(π/4 - α/2), tanα + tanβ = sin(α+β) / cosαcosβ, cotα + cotβ = sin(α+β) / sinαsinβ, tanα + cotα = 2 / sin2α. Difference of trigonometric: sinα - sinβ = 2sin((α-β)/2)cos((α+β)/2), cosα - cosβ = -2sin((α+β)/2)sin((α-β)/2), cosα - sinα = (square root of 2)cos(π/4+α) = (square root of 2)sin(π/4-α), 1 - cosα = 2sin^2(α/2), 1 - sinα = 2cos^2(π/4 + α/2) = 2sin^2(π/4 - α/2), tanα - tanβ = sin(α-β) / cosαcosβ, cotα - cotβ = sin(β-α) / sinαsinβ, cotα - tanα = 2cot2α.