All types of list of trigonometric identities

We can categorize trig identities in many form, Below is a list of trigonometric identities,

Pythagoras trigonometric identities

  1. sin2x + cos2x = 1

  2. 1 + tan2x = sec2x

  3. 1 + cot2x = cosec2x

Reciprocal trigonometry identities

  1. sinA = 1 / cosecA

  2. cosA = 1 / secA

  3. tanA = 1 / cotA

Quotient trig identities

  1. tanA = sinA / cosA

  2. cotA = cosA / sinA

double angle trig identities

  1. sin2A = 2sinAcosA

  2. cos2A = cos2A – sin2A = 2cos2A – 1 = 1 – 2sin2A

  3. tan2A = 2tanA / (1 – tan2A)

Half angle identities
Half angle identities

Complex Trigonometric Identities

Complex Trigonometric Identities

Inverse trigonometric identities


  1. sin-1x + cos-1x = π / 2
  2. tan-1x + cot-1x = π / 2
  3. sec-1x + cosec-1x = π / 2

All above fundamental trigonometric identities can be identified as trigonometric identities table. Now let us enhance our knowledge by proving trigonometric identities. Here is a trigonometric identities proof for some of trig identities.

By using Pythagoras theorem in above right-angle triangle ABC,
right-angle triangle

AC2 = AB2 + BC2 -------------- (1)

=> 1 = (AB2 / AC2) + (BC2 / AC2)

=> 1 = sin2C + cos2C (as, sinC = AB/AC and cosC = BC/AC)

=> sin2C + cos2C = 1

for second trigonometric identity divide equation (1) by AB2,

so, (AC2 / AB2) = (AB2 / AB2) + (BC2 / AB2)

=> (AC2 / AB2) = 1 + (BC2 / AB2)

=> cosec2C = 1 + cot2C

=> 1 + cot2C = cosec2C

for third divide equation (1) by BC2,

so, (AC2 / BC2) = (AB2 / BC2) + (BC2 / BC2)

=> (AC2 / BC2) = (AB2 / BC2) + 1

=> sec2C = tan2C + 1

=> tan2C + 1 = sec2C

In the next section we will see procedure for verifying trigonometric identities.

Process of verifying trig identities

Currently you might have a question how to verify trigonometric identities? Let us see procedure for verifying trig identities by example.
verifying trig identities

Now we will see difference between trigonometric identities and equations. Trigonometric identities help us to change one trig function to another where as in case of equations we have to find values of angles which makes trig statement true. Let's have a example for that,
trigonometric identities and equations


Up till now we have finished almost all trigonometric identities, now its time to find out how to solve trigonometric identities. But before solving trigonometric identities we will first see derivatives of trigonometric functions.
derivatives of trigonometric functions

All above identities will help you to solve trigonometric identities problems, let us check how.
solve trigonometric identities problems

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