Trig Identities

1. Trig Identities on the six trigonometric ratios:

(Note: trig is an abbreviation for trigonometric. We will use trig for the word trigonometric to avoid repetition and therefore boredom.)


triangle with sin, cos, tan meaning


  1. Sinα = opposite side/hypotenuse = AB/AC
  2. Cosα = adjacent side/hypotenuse = BC/AC
  3. Tanα = opposite side/adjacent side = AB/BC

2. The reciprocal Trig identities

  1. cosecα = 1/sinα
  2. secα = 1/cosα
  3. cotα = 1/tanα

3. The three Pythagorean trig identities

  • Sin2α + Cos2α = 1
  • Sec2α – tan2α = 1, Sec2α = 1+ tan2α
  • Cosec2α – cotan2α = 1, Cosec2α = 1+ cotan2α

4. The quotient Trig identities:

  • tan α = sin α /cos α
  • cot α = cos α /sin α

5. Trig identities of negative angles:

  • Sin (– α) = –sin α
  • cos (– α) = cos α
  • tan (– α) = – tanα
  • cot (– α) = – cotα
  • Sec (– α) = Sec α
  • cosec (– α) = – cosec α

6. Trig identities of compound angles

  • sin (A + B) = sinAcosB + cosAsinB
  • sin (A – B) = sinAcosB – cosAsinB
  • cos (A + B) = cosAcosB – sinAsinB
  • cos (A – B) = cosAcosB + sinAsinB
  • Tan(A + B) = (sinAcosB + cosAsinB)/(cosAcosB – sinAsinB)
  • Tan(A – B) = (sinAcosB – cosAsinB)/(cosAcosB + sinAsinB)

7. Trig identities on products of compound angles

  1. sin(A + B)sin(A – B) = sin2A – sin2B Or cos2B – cos2A
  2. cos (A + B)cos(A – B) = cos2A – sin2B Or cos2B – sin2A
  3. tan (A + B)tan(A – B) = (tan2A – tan2B)/ (1 – tan2Atan2B)
  4. tan(A + B + C) = (tan A + tan B + tan C – tanA tanB tanC)/ (1– tanAtanB – tanB tanC –tanC tanA)

8. Trig identities on conversion of sum of angles to product of angles

  • sin (A + B) + sin ( A – B) = 2sinAcosB
  • sin (A + B) – sin (A – B) = 2 cosAsinB
  • cos ( A + B ) + cos (A – B) = 2cosAcosB
  • cos ( A – B ) – cos (A + B) = 2sinAsinB

9. Trig identities on sums of angles.

10. trig identities on double angles:

  • Sin2A = 2sinAcosA Or 2tanA/ (1 + tan2A)
  • cos2A = 2cos2A – 1 Or 1 – 2sin2A Or (1 – tan2A)/(1 + tan2A)
  • tan2A = 2tanA/(1 – tan2A)

11. Trig identities on triple angles

  • Sin3A = 3sinA – 4sin3A
  • cos3A = 4cos3A – 3cosA
  • tan3A = (3tanA – tan3A)/(1 – 3tan2A)

12. Trig identities on sub-multiple angles

  • sin2A + sin2B + sin2C = 4sinAsinBsinC Or -1 – 4cosAcosBcosC
  • cos2A + cos2B + cos2C = 1 – 4 sinAsinBsinC
  • tan2A + tan2B + tan2C = tan2C tan2B tan2C

13. The law of sines Or the sine rule:

14. the law of cosines or the cosine rule

  • a2 = b2 + c2 – 2bc cosA
  • b2 = a2 + c2 – 2ac cosB
  • c2 = b2 + a2 – 2abcosC

15. the projection formula

  • a = bcosC + ccosB
  • b = acosC + ccosA
  • c = bcosA + acosB