• 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(π2−x)=sinx
Identity 6: sin(π2−x)=cosx
Identity 7: sin(x+y)=sinxcosy+cosxsiny
Identity 8: sin(x−y)=sinxcosy−cosxsiny
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+tany1−tanxtany
Identity 11: tan(x−y)=tanx−tany1+tanxtany
Identity 12: cot(x+y)=cotxcoty−1coty+cotx
Identity 13: cot(x−y)=cotxcoty+1coty−cotx
Identity 14:
cos2x=cos2x−sin2x=1−2sin2x=2cos2x−1=1−tan2x1+tan2x
Identity 15: sin2x=2sinxcosx=2tanx1+tan2x
Identity 16: tan2x=2tanx1−tan2x
Identity 17: sin 3x = 3 sin x - 4 sin3x
Identity 18: cos 3x = 4 cos3x - 3 cos x
Identity 19: tan3x=3tanx−tan3x1−3tan2
Identity 20(a): cosx+cosy=2cosx+y2cosx−y2
Identity 20(b): cosx−cosy=−2sinx+y2sinx−y2
Identity 20(c): sinx+siny=2sinx+y2cosx−y2
Identity 20(d): sinx−siny=2cosx+y2sinx−y2
Identity 21(a): 2cosxcosy=cos(x+y)+cos(x−y)
Identity 21(b): −2sinxsiny=cos(x+y)−cos(x−y)
Identity 21(c): 2sinxcosy=sin(x+y)+sin(x−y)
Identity 21(d): 2cosxsiny=sin(x+y)−sin(x−y)
Identity 22(a): sinx = ±√1−cos2x2
Identity 22(b): sinx2 = ±√1−cosx2
Identity 23(a): cosx = ±√1+cos2x2
Identity 23(b): cosx2 = ±√1+cosx2
Identity 24(a): tanx = 1−cos2xsin2x
Identity 24(b): tanx = sin2x1+cos2x
Identity 24(c): tanx2 = 1−cosxsinx
Identity 24(d): tanx2 = sinx1+cosx
Solved examples related to the above identities can be seen here.
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