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Thursday, January 6, 2022

List of Trigonometric Identities

• Given below is the list of 24 trigonometric identities.
• Derivation of the first nine identities can be seen here.
• Derivation of the remaining identities can be seen here.

Identity 1: sin (-x) = -sin x  

Identity 2: cos (-x) = cos x 

Identity 3: cos (x+y) = cos x cos y - sin x sin y 

Identity 4: cos (x-y) = cos x cos y + sin x sin y 

Identity 5: cos(π2x)=sinx

Identity 6: sin(π2x)=cosx

Identity 7: sin(x+y)=sinxcosy+cosxsiny

Identity 8: sin(xy)=sinxcosycosxsiny

Identity 9(a): cos(π2+x)=sinx

Identity 9(b): sin(π2+x)=cosx 

Identity 9(c): cos (π - x) = - cos x 

Identity 9(d): sin (π - x) = sin x 

Identity 9(e): cos (π + x) = -cos x  

Identity 9(f): sin (π + x) = -sin x 

Identity 9(g): cos (2π - x) = cos x.

Identity 9(h): sin (2π - x) = -sin x 

Identity 10: tan(x+y)=tanx+tany1tanxtany

Identity 11: tan(xy)=tanxtany1+tanxtany

Identity 12: cot(x+y)=cotxcoty1coty+cotx

Identity 13: cot(xy)=cotxcoty+1cotycotx

Identity 14:
cos2x=cos2xsin2x=12sin2x=2cos2x1=1tan2x1+tan2x

Identity 15: sin2x=2sinxcosx=2tanx1+tan2x

Identity 16: tan2x=2tanx1tan2x 

Identity 17: sin 3x = 3 sin x  - 4 sin3x

Identity 18: cos 3x = 4 cos3x - 3 cos x

Identity 19: tan3x=3tanxtan3x13tan2

Identity 20(a): cosx+cosy=2cosx+y2cosxy2

Identity 20(b): cosxcosy=2sinx+y2sinxy2

Identity 20(c): sinx+siny=2sinx+y2cosxy2 

Identity 20(d): sinxsiny=2cosx+y2sinxy2

Identity 21(a): 2cosxcosy=cos(x+y)+cos(xy)

Identity 21(b): 2sinxsiny=cos(x+y)cos(xy)

Identity 21(c): 2sinxcosy=sin(x+y)+sin(xy)

Identity 21(d): 2cosxsiny=sin(x+y)sin(xy)

Identity 22(a): sinx = ±1cos2x2

Identity 22(b): sinx2 = ±1cosx2

Identity 23(a): cosx = ±1+cos2x2

Identity 23(b)
: cosx2 = ±1+cosx2

Identity 24(a): tanx = 1cos2xsin2x

Identity 24(b): tanx = sin2x1+cos2x

Identity 24(c): tanx2 = 1cosxsinx

Identity 24(d): tanx2 = sinx1+cosx


Solved examples related to the above identities can be seen here.

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