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Trigonometric Identities

credit: Shawna Haider

Remember an identity is an equation that is true for all defined values of a

variable.

The left-hand expression always equals the right-hand expression, no matter what x equals.

xxx 2

RECIPROCAL IDENTITIES

sin

1cosec

cos

1sec

tan

1cot

QUOTIENT IDENTITIES

cos

sintan

sin

coscot

22 sec 1tan

22 cosec cot1

PYTHAGOREAN IDENTITIES

1 cossin 22

EVEN-ODD IDENTITIES

cotcotsecseccoseccosec

tantancoscossinsin

Note:

In every identities we can write every functions in terms of sine and cosine.

We can create different versions of many of these identities by using arithmetic.

22 sincoscosecsin Establish the following identity:

In establishing an identity you should NOT move things from one side of the equal sign to the other. Instead substitute using identities you know and simplifying on one side or the other side or both until both sides match.

22 sincoscosecsin Let's sub in here using reciprocal identity

22 sincossin

1sin

22 sincos1

We often use the Pythagorean Identities solved for either sin2 or cos2.

sin2 + cos2 = 1 solved for sin2 is sin2 = 1 - cos2 which is our left-hand side so we can substitute.

22 sinsin

We are done! We've shown the LHS equals the

RHS

cos1

sincotcosec

Establish the following identity:

Let's sub in here using reciprocal identity and quotient identity

Another trick if the denominator is two terms with one term a 1 and the other a sine or cosine, multiply top and bottom of the fraction by the conjugate and then you'll be able to use the Pythagorean Identity on the bottom

We worked on LHS and then RHS but never moved things

across the = sign

cos1

sincotcosec

cos1

sin

sin

cos

sin

1

cos1

sin

sin

cos1

combine fractions

cos1

cos1

cos1

sin

sin

cos1

2cos1

cos1sin

sin

cos1

FOIL denominator

2sin

cos1sin

sin

cos1

sin

cos1

sin

cos1

Get common denominators

If you have squared functions look for Pythagorean Identities

Work on the more complex side first

If you have a denominator of 1 + trig function try multiplying top & bottom by conjugate and use Pythagorean Identity

When all else fails write everything in terms of sines and cosines using reciprocal and quotient identities

Have fun with these---it's like a puzzle, you can use identities and algebra to get them to match!

Hints for Establishing Identities

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