Calculus 2 - Trigonometric Identities

Reciprocal identities

sinx=1/cscx, cscx=1/sinx, cosx=1/secx, secx=1/cosx, tanx=1/cotx, cotx=1/tanx

Quotient identities

tanx=sinx/cosx, cotx=cosx/sinx

Pythagorean identities

sin²x+cos²x=1, tan²+1=sec²x, cot²x+1=csc²x

Cofunction identities

sinx=cos((pi/2)-x), cosx=sin((pi/2)-x), cscx=sec((pi/2)-x), secx=csc((pi/2)-x), tanx=cot((pi/2)-x), cotx=tan((pi/2)-x)

Even-odd identities

sin(-x)=-sinx, cos(-x)=cosx, csc(-x)=-cscx, sec(-x)=secx, tan(-x)=-tanx, cot(-x)=-cotx

Sum of angles identities

sin(A+B)=sinAcosB+cosAsinB, cos(A+B)=cosAcosB-sinAsinB, tan(A+B)=(tanA+tanB)/(1-tanAtanB)

Difference of angles identities

sin(A-B)=sinAcosB-cosAsinB, cos(A-B)=cosAcosB+sinAsinB, tan(A-B)=(tanA-tanB)/(1+tanAtanB)

Power-reducing identities

sin²x=(1-cos2x)/2, cos²x=(1+cos2x)/2, tan²x=(1-cos2x)/(1+cos2x)

Double-angle identities

sin2x=2sinxcosx, cos2x=cos²x-sin²x, cos2x=1-2sin²x, cos2x=2cos²x-1, tan2x=(2tanx)/(1-tan²x)

Half-angle identities

sin(x/2)=±sqrt((1-cosx)/2), cos(x/2)=±sqrt((1+cosx)/2), tan(x/2)=±sqrt((1-cosx)/(1+cosx)), tan(x/2)=(1-cosx)/sinx, tan(x/2)=sinx/(1+cosx)

Product-to-sum identities

sinAsinB=(1/2)[cos(A-B)-cos(A+B)], cosAcosB=(1/2)[cos(A-B)+cos(A+B)], sinAcosB=(1/2)[sin(A+B)+sin(A-B)], cosAsinB=(1/2)[sin(A+B)-sin(A-B)]

Sum-to-product identities

sinA+sinB=2sin((A+B)/2)cos((A-B)/2), cosA+cosB=2cos((A+B)/2)cos((A-B)/2), sinA-sinB=2cos((A+B)/2)sin((A-B)/2), cosA-cosB=-2sin((A+B)/2)sin((A-B)/2)