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Proving Trigonometric Identities

Quick Review:

22

22

22

csc1cot

sectan1

1cossin

Quotient Identities

Reciprocal Identities Pythagorean Identities

We Verify (or Prove) Identities by doing the following:

Work with one side at a time.We want both sides to be exactly the

same.Start with either sideUse algebraic manipulations and/or the

basic trigonometric identities until you have the same expression as on the other side.

Change everything on both sides tosine and cosine.

Suggestions

Start with the more complicated side Try substituting basic identities (changing all functions to

be in terms of sine and cosine may make things easier) Try algebra: factor, multiply, add, simplify, split up

fractions If you’re really stuck make sure to:

Remember to:

• Work with only one side at a time!

xx

x

xx

cotsin

1cos

csccos RHS

xxxxxx 2sincostancoscotsin

Let’s start by working on the left side of the equation….

xxxxxx 2sincostancoscotsin

x

xx

x

xx

cos

sincos

sin

cossin

Rewrite the terms inside the second parenthesis by using the quotient identities

xxxxxx 2sincostancoscotsin

x

xx

x

xx

cos

sincos

sin

cossin

Simplify

xxxxxx 2sincostancoscotsin

x

xx

x

xx

sin

sin

1

sin

sin

cossin

To add the fractions inside the parenthesis, you must multiply by one to get common denominators

xxxxxx 2sincostancoscotsin

x

x

x

xx

sin

sin

sin

cossin

2

Now that you have the common denominators, add the numerators

xxxxxx 2sincostancoscotsin

x

xxx

sin

sincossin

2

Simplify

xxxxxx 2sincostancoscotsin

xxxx 22 sincossincos

Since the left side of the equation is the same as the right side, you’ve successfully proven the identity!

On to the next problem….

xxxx 2244 sincossincos

Let’s start by working on the left side of the equation….

xxxx 2244 sincossincos

xxxx 2222 sincossincos

We’ll factor the terms using the difference of two perfect squares technique

xxxx 2244 sincossincos

1sincos 22 xx

Using the Pythagorean Identities the second set of parenthesis can be simplified

xxxx 2244 sincossincos

xxxx 2222 sincossincos

Since the left side of the equation is the same as the right side, you’ve successfully proven the identity!

On to the next problem….

x

xxx

sin1

cossectan

Let’s start by working on the right side of the equation….

x

xxx

sin1

cossectan

x

x

x

x

sin1

sin1

sin1

cos

Multiply by 1 in the form of the conjugate of the denominator

x

xxx

sin1

cossectan

x

xx2sin1

)sin1(cos

Now, let’s distribute in the numerator….

x

xxx

sin1

cossectan

x

xxx2cos

sincoscos

… and simplify the denominator

x

xxx

sin1

cossectan

x

xx

x

x22 cos

sincos

cos

cos

‘Split’ the fraction and

simplify

x

xxx

sin1

cossectan

x

x

x cos

sin

cos

1

Use the Quotient and Reciprocal Identities to rewrite the fractions

x

xxx

sin1

cossectan

xx tansec

And then by using the commutative property of addition…

x

xxx

sin1

cossectan

xxxx sectansectan

… you’ve successfully proven the identity!

One more….

xxx

2csc2cos1

1

cos1

1

Let’s work on the left side of the equation…

xxx

2csc2cos1

1

cos1

1

x

x

xxx

x

cos1

cos1

cos1

1

cos1

1

cos1

cos1

Multiply each fraction by one to get the LCD

xxx

2csc2cos1

1

cos1

1

x

x

x

x22 cos1

cos1

cos1

cos1

Now that the fractions have a common denominator, you can add the numerators

xxx

2csc2cos1

1

cos1

1

x

xx2cos1

cos1cos1

Simplify the numerator

xxx

2csc2cos1

1

cos1

1

x2cos1

2

Use the Pythagorean Identity to rewrite the denominator

xxx

2csc2cos1

1

cos1

1

x2sin

12

Multiply the fraction by the constant

xxx

2csc2cos1

1

cos1

1

xx 22 csc2csc2

Use the Reciprocal Identity to rewrite the fraction to equal the expression on the right side of the equation

How to get proficient at verifying identities:

Once you have solved an identity go back to it, redo the verification without looking at how you did it before, this will make you more comfortable with the steps you should take.

Redo the examples done in class using the same approach, this will help you build confidence in your instincts!

Don’t Get Discouraged!

Every identity is differentKeep trying different approachesThe more you practice, the easier it will be

to figure out efficient techniquesIf a solution eludes you at first, sleep on it!

Try again the next day. Don’t give up!You will succeed!

Establish the identity

Establish the identity

Establish the identity

Homework

TBA

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