Throughout this section you will learn basic Trigonometric Identities: Reciprocal Identities, Pythagorean Identities and Quotient Identities. By learning the three above noted identities, you will then learn how to Prove Identities. Every topic links together to make it more comprehensible. Trigonometric Identities are just a section of Trigonometry. The Law of Cosines, the Law of Sines, the Pythagorean Theorem and Right Angle Triangles each play a part in learning identities.

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TRIGONOMETRYDiscovered in the 17th century and is derived from the Latin

word trigonometria. This is defined as the study of the Triangle and of the relationships between the angles and sides of a triangle, plus the deduction of certain components of the triangle when others are known.

TRIGONOMETRIC IDENTITES This comes from the basics of trigonometry, dealing with the

relations of the sides and angles of triangles. Each trigonometric identity has a reciprocal identity.

IDENTITIESDiscovered in the 16th century and comes from the Latin word

Identitas, which are equations that are true for all values of the variable for which they are defined.

Now that we know the definition for “Trigonometr(y)/ (ic) Identities”, we can move onto trigonometric functions. For most trigonometric functions the three functions, Sine, Cosine and Tangent are used most frequently. Throughout this unit of Trigonometric Identities, three more functions are added, Cotangent, Secant and Cosecant. There are six ways of making ratios of two sides of a right angle. So, remember each identity has a reciprocal identity. As long as you can remember that, then learning the identities will not be difficult. a chart of the six functions and their values.

Identity Letter from Triangle Position on Triangle

1. SinA= a/c Opposite/ Hypotenuse2. CosA= b/c Adjacent/ Hypotenuse3. TanA= a/b Opposite/ Adjacent4. CotA= b/a Adjacent/ Opposite5. SecA= c/b Hypotenuse/ Adjacent6. CscA= c/a Hypotenuse/ Opposite

All reciprocal identities are, are the six identities and what each one equals. Once you remember one, the other is just the opposite. Below we have a chart of the reciprocal identities.

Sinx= 1 Reciprocal Cscx= 1 cscx sinx

Cosx= 1 Reciprocal Sec= 1 secx cosx

Tanx= 1 Reciprocal Cotx= 1 Cotx tanx

Do you see the similarities? Congratulations, these are the Reciprocal Identities.

Next are Quotient Identities. All the identities are based on Sine and Cosine. These can come in handy when proving identities.Below are the two quotient identities. These identities have two meanings each. One, the quotient identity and the other, the reciprocal identities.

Tanx= sinx Cotx= cosx cosx sinx

The following are the Pythagorean Identities, which come from the Pythagorean Theorem. The Pythagorean Theorem is solving right angle triangles and an unknown side. This formula looks as:

a2 + b2 = c2

Now if you look back to the chart of the six functions and their values you can see from the triangle where “a”, “b” and “c” come from. “C” in this equation is equal to the hypotenuse. This then translates very easily into a Pythagorean Identity for sines and cosines. Divide both sides by c2 and you will receive….

a2/ c2 + b2/ c2 = 1

a2/ c2 = Sin2x and b2/ c2 = Cos2x.

The Pythagorean Identity for sines and cosines is:

sin2x + cos2x = 1Therefore to figure out the next two Pythagorean Identities is very easy. Simply divide each term by a new letter rather than just “c”. You will divide by “a” and “b”. When dividing a letter by itself, it then equals one.

Here goes:

a2 + b2 = c2

a2/ a2 = 1 b2/ a2 = cot2x c2/ a2 = csc2x

Now if you look in the chart of the six functions, you see how we came up with naming b/a and c/a. Look under “Letter from Triangle”.

The Pythagorean Identity for cotangent and cosecant is:

1 + cot2x = csc2x

Now repeat for the letter B.

a2 + b2 = c2

a2/ b2 = tan2x b2/ b2 = 1 c2/ b2 = sec2x

The Pythagorean Identity for tangent and secant is:

1 + tan2x = sec2x

Now you have completed the Pythagorean Identities.

In the above equations “x” is a variable. When using a value for x, it can be in either degrees or radians for the equation to hold true.

Now these degrees or radians can come from the Unit Circle which is the fundamental ideas needed to understand sines and cosines.

Example This is related to the Pythagorean Theorem.

Sinx =y =y Cosx = x 1 1 Now we know that sin2 x + cos2 x = 1 This is where the Pythagorean Theorem comes into play. You will recognize this equation of Pythagorean Identity for cosine and sine.

Now this is where things could get tricky. Solving all the different equations. Don’t worry as long as you read each step carefully and take your time you’ll be fine. Okay, let’s begin.

Recipricol Identities

You should remember what these are, if not, just scroll back up to the beginning where there is an explanation.

Start o with the easier explanations then work your way up to the harder ones.

Sinx= 1 Reciprocal Cscx= 1 cscx sinx

Cosx= 1 Reciprocal Sec= 1 secx cosx

Tanx= 1 Reciprocal Cotx= 1 cotx tanx

Examples

1. Express each of the following terms into sinx or cosx or both

a) 1 = cosx secx

Looking at this, you can look at the chart and see that the recipricol identity is cosx

b) sin2 x + 1 = sin2 x + cos2 x = 1

This equation involves a little bit of the Pythagorean Identity in the end. First of all, you don’t have to change sin2 x because we want the term to be in sinx or cosx or both. Now, you know that sec2 x is equal to cos2 x. If it is originally squared, leave it squared. Now, you are left with sin x + cos x, this is where Pythagorean comes into play. If you look at the chart, sin2 x + cos2 x = 1. One is the final answer.

c) tanx cosx = sinx cosx = sinx cosx

Now, with this equation, a quotient identity is involved. Can you guess which one? Yes, tan. You will use the quotient identity because you are trying to work with cosines and sines. The usual reciprocal for tan equal 1 but using

cotxquotient identities it is equal to sinx . Then you are left

cosxwith sinx cosx. You are dealing with multiplying this time. cosxBefore you start multiplying, you will see that you can cancel out the two cosines, because when multiplying there are identical terms that will cancel out. Now, you are left with just sinx.

d) csc2 x = 1 / cos2 x = 1 * sin2 x = 1 cot2 x sin2 x sin2 x sin2 x cos2 x cos2 x

Now, it gets a little more confusing. Again, we are having to use a quotient identity. Now the reciprocal of csc2 x = 1 sin2 xRemember to leave it squared. Next cot2 x = cos2 x.

sin2 xthis is the quotient identity. When dividing, you take the second term and flip it and now change the dividing sign to a multiplication sign. Now, you can cancel out the sin x and we are left with 1 cos2 x

e) sec2 x – tan2 x = 1 - sin2 x =1 – sin2x = cos2 x = 1 cos x cos2 x cos2 x cos2 x

This equation involves Pythagorean and quotient as well as reciprocal. The reciprocal to sec x is 1 and the quotient

cos2 x identity is sin2 x. Now when subtracting, you work straight cos2 xacross, 1 – sin2 x equals 1 – sin2 x. The denominator for this equation is cos2 x because you leave the denominator the same when the two terms are identical. Now as you look at Pythagorean Identities, you can see that 1 – sin2 x equals cos2 x. You have to rearrange the equation. You must bring sin2 x over to the side of one. When switching sides, you switch signs. We are now left with cos2 x whichequals one.

cos2

x

Wow! You have just completed reciprocal identities with a little of the other identities involved.

Next is Quotient Identities and if you don’t understand what these are, just scroll back up to the beginning and read the explanation.

Quotient Identities

tanx = sinx cotx = cosx cosx sinx

So here you are only dealing with two identities, therefore you will see these identities combined with the other ones.

Examples

1. Each of the following can be expressed in terms of sinx or cosx or both.

a) cot2 x = cos2 x sin2 x

If you look at the chart, you will see that cot x is equal to, cos2 x

sin2 x

remember to keep them squared.

b) 1 + cot2 x = csc2 x = 1 / cos2 x = 1 * sin2 x = 1 cot2 x cot2 x sin2 x sin2 x sin2 x cos2 x cos2 x

Here you are dealing with the quotient identity and also the reciprocal identity and the Pythagorean Identity. If you look at the chart, you will see that 1 + cot2 x is equal to csc2 x, which is also equal to 1 in the reciprocal identities. sin2 xRemember that cot2x is equal to cos2x. sin2 xFlip the second term and switch the dividing sign to a multiplying sign. Also when you have identical terms that

are opposite to each other, you can cancel them out. Now you are left with 1 cos2 x

c) 1 + tanx = 1 + sinx / 1 = cosx + sinx * cosx = cosx+sinx secx cosx cosx cosx 1

The quotient identity of tanx is sinx. cosx The reciprocal identity of secx is 1 cosxAgain, you can flip the second term and switch the dividing sign into a multiplication sign. The two cosx cancel out because they are opposites. Now you are left with cosx + sinx.

d) Write each of the following terms of cosx.

cotx sinx = cosx sinx = cosx sinx

The quotient identity for cotx is cosx sinx Leave sinx the same so you can cancel the two signs out because they are opposite (one on top and one on bottom). When they are cancelled, you are left with cosx.

Now we will move onto the Pythagorean Identities. You can rewrite each Pythagorean Identity by changing the signs and the terms to the opposite sides.

Pythagorean Identities

sin2 x + cos2 x = 1

1 + tan2 x = sec2 x

1 + cot2 x = csc2 x

Examples

An example of changing the equation around

sin2 x + cos2 x = 1 (which can also be written as) 1 – sin2 x = cos2 x

What was done here was to move sin2 x onto the other side. The sign was changed to a negative.

Now examples of equations:

1. Express each of the following in terms of sinx or cosx or both

a) 1 – csc2 x = cot2 x

To find out what 1 – csc2 x equals, you look at the chart of Pythagorean Identities and for this one you must rearrange the equation 1 + cot2 x = csc2 x.

b) tanx = sinx 1 + tanx cosx + sinx

sinx / cosx + sinx = sinx * cosx = sinx

cosx cosx cosx cosx + sinx cosx + sinx

For this equation, you found the quotient identity for tanx that equals sin2 x You must also use quotient identities cos2 x c) cotx = cosx / sinx + cosx = cosx * sinx = cosx 1 + cotx sinx sinx sinx sinx+cosx sinx+cosx

This equation resembles the one above. You are again dividing with quotient identities. You found the quotient identity for cotx which equals cosx and found 1 + tanx

sinx

which is equal to cosx and one equals sinx because of the sinxdenominator. Again take the second term and flip it and change the division sign to a multiplication sign Then

cancel the sinx out because of the opposite sides. You are now left with cosx sinx + cosx

How To Prove Identities

You can now begin to prove Identities. Proving identities is a variety of the three previously mentioned identities. There is not a specific formula but rather you work with the reciprocal identities, the Pythagorean theorem and the quotient identities to change the terms around so you end up with two identical terms. Proving Identities can be a little difficult but once you get the hang of it, it will be very easy. Go through each step one at a time and analyse the term(s) carefully. If you have, any problems don’t forget to look at the different charts to find helpful hints.

Examples

1. cosx cscx = cotx

cosx 1 = cosx sinx sinx

cosx = cosxsinx sinx

What was done was to show that cosx cscx was equal to cotx. Then the reciprocal for cscx which is 1 was found. sinxThe quotient identity for cotx was used. Then cosx was multiplied by 1 which was equal to cosx over sinx. The two equations were then equal.

2. cosx - sinx = sinx – 1cscx tanx secx

cosx / 1 - sin / sinx = sinx – 1 sinx cosx secx

cosx sinx - sinx cosx = sinx – 1 1 sinx secx

cosx sinx - cos = sinx – 1 secx

sinx - 1 = sinx – 1secx secx secx

sinx – 1 = sinx – 1 secx secx

You began with finding the reciprocal to cscx which is 1 . sinx

Next, find the quotient identity for tanx which is sinx cosx

Then flip the second term in each equation. Change the division signs to multiplication signs. Then multiply cosx by sinx and you are left with cosx sinx. When looking at the next terms, you can see that you can cancel out the sinx’s because they are opposites. Now you are left with cosx sinx – cosx. For each cosx, use the reciprocal identity 1 When multiplying sinx by 1, you get sinx. You are

secx

then left with sinx - 1 . Then when you subtract, you are secx secxleft with sinx – 1. Always leave the denominator. The two sides secxare equal.

3.(1 – secx) (1 + secx) = - tan2 x

1 – sec2 x = - tan2 x

- tan2 x = - tan2 x

For this example, you are expanding using difference of squares:1 x 1 = 1; 1 x secx = secx; - secx x 1 = -secx. The two secx cancel.- secx x secx = sec x. You are now left with 1 – sec x which, if

you refer to the Pythagorean Identities, is equal to – tan x.

3. six2 x cot2 x + cos2 x = 1

cot2 x

sin2 x cos2 x + cos2 x cos2 x = 1 sin2 x sin2 x = 1

cos 2x + cos2 x * sin2 x = 1 cos2 x

cos x + sin2 x =1 1=1

What was done here was to find the quotient identity for cot2 x, which is cos 2 x . The first section of the equation was

sin2 xmultiplied therefore you could cancel out the two sin x because they were opposite. Now, for the second part of the equation. You flip the second terms and then you can cancel out the two cos2 x because they are opposite. Now you are left with cos2 x + sin2 x which if you look at the Pythagorean Theorem chart you see that this is equal to 1.1 is the final answer.

Trig Identity ExamplesProve each of the following Identities

1.tan x cos = sinxsinx/cos x cos = sinxsinx = sinx

2.secx +1/cotx =1+sinx/cosx1/cosx+1/cotx=1+sinx/cosx1+sinx/cosx=1+sinx/cosx

3.secx/sinx=1/sinx+1/cosxsecx/sinx=1/sinx+secx/1secx/sinx=secx/sinx

4.

cos x cscx=cotxcosx/1 x 1/sinx=cotxcosx/sinx=cotxcotx = cotx

5. (1+cosx)(1-cosx)=sin2x1-cosx+cosx -cos2x = sin2

1-cos2x = sin2x sin2x = sin2

6.sinx(sin2x + csc x cos2)=1sinx(sin2x + cosx/sinx)=1sin2x + cos2x =11=1

7.sin2 x sec x cscx=tansin2x/1 x 1/cos x 1/sinx=tanxsinx/cosx=tanxtanx=tanx

8.(1-secx)(1+secx)=-tan2x

1-sec2x = -tan2x 1-1/cos2x = -tan2

cos2x/cos2x -1/cos2x = -tan2

-sin2/cos2=-tan2

-tan2x = -tan2

9.1-tanx/tanx =cosx-sinx/sinx1/tan -1=cosx-sinx/sinx1/sinx/cosx-1=cosx-sinx/sinxcox/sinx-sinx/sinx=cosx-sinx/sinxcosx-sinx/sinx=cosx-sinx/sinx

10.sin3 x cos2 x = cos2 x sinx - cos4 x sinxsin3 x cos2x =cos2 x sinx (1-cos2x)sin3 x cos2x = sin2x sin x sin2x

sin3 x cos2x = sin3 x cos2x11.

secx/cotx = sinx/cos2

1/cos x sinx/cosx = sinx/cos2

sinx/cos2x = sinx/cos2x12.

sinx/cosx sinx/cosx=tan2

sin2x/cos2 x = tan2

tan2=tan2

13.(cosx/secx-1)-cosx/tan2

(cosx/secx-1)-cosx/(secx-1)(secx+1)=cosx/secx-1 x secx+1/secx+1cos x secx+cosx-cosx/tan2

1/tan2x =cot2

cot2=cot2

14.sins/cosx x cosx/sinx +1= 21+1=22=2

15.1/(sin x cosx)-cosx/sinx=tanx1-cos2x/(sin x cosx)=tanxsin2x/(sin x cosx)=tanxtanx=tanx

Quiz QuestionsProve the following identities

1.(secx/sinx)-(sinx/cosx)=cotx

2.1/(1+cosx)=csc2x-cscx cotx

3.

csc2x-cos2x csc2x=1

4.cot2 csc2 –cot2=cot4

5.(secx+1)(secx-1)=tan2x

Web Sites For Further Help With Trigonometric Identities

1. http://hrsbstaff.ednet.ns.ca Go to “W”. Joan Wells and go into her website.

2. http://mathforum.org/dr.math/ Go to High School then go to Trigonometry. Then scroll down the page to whatever you need help with.

3. http://library.thinkquest.org/17119/ Search throughout this unit and it will show you how to prove Identities.