# FUNDAMENTAL TRIGONOMETRIC IDENTITIES

Reciprocal Identities

Quotient Identities

Pythagorean Identities

**Problem 1:
**Add or subtract the following trigonometric expressions:

(a)

Since both terms have a **sin x **factor, we only have to add the
coefficients.

(b)

Since both terms have a **sec x **factor, we only have to subtract the

coefficients.

(c)

Here we combine the terms containing **cos x** and the terms containing **
tan x**.

(d)

the common denominator is

to add the two fractions the number **1** has to be
multiplied by **cos x** and the

expression **tan x** by **sin x,** and finally we get

(e)

both fractions already have the same denominator, therefore

(f)

the common denominator is

to add the two fractions, the number **3** has to be multiplied by **(tan x
- sec x)**

and the number **5** by (**tan x + sec x)
**

and multiplying out the numerator and combining like terms, we finally get

**Problem 2:**

Multiply the following trigonometric expressions :

(a)

Now we will use FOIL to expand as follows :

(b)

Here we will multiply the coefficients and the trigonometric ratios to get

(c)

Here we will multiply the coefficients and the trigonometric ratios to get

**Problem 3:**

Factor the following trigonometric expressions:

(a)

Notice that every term contains a factor of sin x which
can be factored out as

follows:

(b)

Notice that we are dealing with a the Difference of
Squares and we can factor

as follows:

(c)

Factor the following expression just like the trinomial , that is,

**Problem 4:
**Change the fraction to two terms and
reduce.

Note: You cannot cancel out **cos x** in the fraction
above. Only factors can

be canceled in rational expressions.

**Problem 5:**

can be reduced to a single number. Find this number.

What could we do?

1. Add or subtract trigonometric expressions? NO

2. Multiply trigonometric expressions? NO

3. Factor trigonometric expressions? NO

4. Separate rational trigonometric expressions? NO

5. Use fundamental identities to rewrite an expression? YES

We can use Reciprocal Identities to rewrite tangent,
cotangent, secant, and

cosecant as follows:

NOTE: While this is certainly a good start, it does not
guarantee

success. We might have to give up and think of something else to

do! If you are wondering if you are ever going to have to use this,

wait until you get to calculus. It is often much easier to reduce a

"complicated" trigonometric expression to a single trigonometric

ratio when working with calculus concepts.

Since we learned in algebra to always simplify complex
fractions, we will

multiply both the numerator and the denominator by the LCD **sin x cos x**
just

like we used to do in algebra.

But before we do this, let's combine the fractions in the
numerator of the

complex fraction as follows:

Next, we will distribute and at the same time reduce just
like we learned in

algebra to get

Now what?

1. Add or subtract trigonometric expressions? NO

2. Multiply trigonometric expressions? NO

3. Factor trigonometric expressions? NO

4. Separate rational trigonometric expressions? NO

5. Use fundamental identities to rewrite an expression? YES

We know that
(Pythagorean Identity), therefore,

can be reduced to the number 1.

**Problem 6:**

** **can be reduced
to a single trigonometric ratio, such as cos(x), sin(x), tan(x),

sec(x), csc(x), or cot(x). Find this ratio.

What could we do?

1. Add or subtract trigonometric expressions? NO

2. Multiply trigonometric expressions? NO

3. Factor trigonometric expressions? NO

4. Separate rational trigonometric expressions? NO

5. Use fundamental identities to rewrite an expression? YES

We can use Reciprocal Identities to rewrite tangent and cotangent as follows:

Since we learned in algebra to always simplify complex
fractions, we will

multiply both the numerator and the denominator by the LCD **sin x cos x**
just

like we used to do in algebra.

Next, we will distribute and at the same time reduce just
like we learned in

algebra to get

Now what?

1. Add or subtract trigonometric expressions? NO

2. Multiply trigonometric expressions? NO

3. Factor trigonometric expressions? YES

4. Separate rational trigonometric expressions? NO

5. Use fundamental identities to rewrite an expression? NO

Let's factor common factors out of the numerator and the
denominator for a

lack of anything better to do.

As you can see, the numerator and denominator have a
factor in common and

when reduced we end up with

. Finally, we do know that this equals .

Therefore, we were able to reduce to the single trigonometric ratio .

**Problem 7:**

can be reduced to a
single trigonometric ratio, such as cos(x), sin

(x), tan(x), sec(x), csc(x), or cot(x). Find this ratio.

First, we will use to
rewrite the expression as .

This is also equal to .

For a lack of anything better to do, let's write the last
expression as a single fraction.

We see that we have the Pythagorean Identity
in the numerator, so

that we can replace it with 1 to get

which equals
.

**Problem 8:**

can be reduced to a
single trigonometric ratio, such as cos

(x), sin(x), tan(x), sec(x), csc(x), or cot(x). Find this ratio.

Recognizing the Pythagorean Identity
in the numerator, we can

change the expression as follows:

which also equals

Lastly, we can cancel out the expression
since it occurs both in the numerator

and in the denominator to find which equals
.

**Problem 9:**

can be reduced to a
difference of two trigonometric ratios. Find this

difference.

What could we do?

2. Multiply trigonometric expressions? NO

3. Factor trigonometric expressions? NO

4. Separate rational trigonometric expressions? NO

5. Use fundamental identities to rewrite an expression? YES

We can use Reciprocal Identities to rewrite secant and cosecant as follows:

Since we learned in algebra to always simplify complex
fractions, we will

multiply both the numerator and the denominator by the LCD **sin x cos x**
just

like we used to do in algebra.

But before we do this, let's combine the fractions in the
numerator of the

complex fraction as follows:

Please note that it is not mandatory to write **sin x **
as the first factor in the

product **sin x cos x**. However, it has become "unofficial" standard
practice

to do so!

Next,

Finally, we will distribute and at the same time reduce
just like we learned in

algebra to get

We find that can be reduced to the difference

.

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