Page 465 from the C+4B Text

Reciprocal Identities Quotient Identities Pythagorean Identities

cscθ =1

sinθ

secθ = 1cosθ

cotθ =1

tanθ, tanθ ≠0

,sinθ ≠0tanθ =

sinθcosθ

,cosθ ≠0

cotθ =cosθsinθ

,sinθ ≠0

sin2θ + cos2θ =1

1+ tan2θ =sin2θ

1+ cot2θ =csc2θ

Let’s start off with an easy example:

(1−sin2 θ)gtan2 θ =sin2 θ

We will make the left side look like the right first by using the Pythagorean identity

(cos2 θ)gtan2 θ =sin2 θ

Next, we will re-write tan using the quotient identity

We will finish by reducing cosine and both sides will now be identical.

sin2 θ =sin2 θ

From the last slide:

Here is another example:

(sin x)(cot x +cosxtanx) =cosx+sin2 x

We will start by working on theleft side of the equation by rewriting the sine and cosine using the quotient identity:

(sin x)cos x

sin x+ cosx*

sinxcosx

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎤

⎦⎥

(sin x)(cot x +cosxtanx) =cosx+sin2 x

(sin x)cos x

sin x+ cosx*

sinxcosx

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎤

⎦⎥

Now we can cross cancel inside the parenthesis

(sin x)cos x

sin x+sinx⎡

⎣⎢⎤⎦⎥

Inside the brackets we need a common denominator

Next, we combine the fraction

(sin x)cos x +sin2 x

sinx⎡

⎣⎢

⎤

⎦⎥

Again we can cross cancel sinx and we are left with…

cos x +sin2 x Which equals the right side!

tan x +secx=cosx

1−sinx

Let’s start by working on the right side of the equation by multiplying by 1 in the conjugate of the denominator.

Here is another example:

Multiply the denominator (hint: use foil)

=cos x(1+ sin x)

1− sin2 x

=cos x(1+ sin x)

cos2 x

Use the Pythagorean indentify to simplify the denominator

Distribute the numerator:

=cos x + cos x sin x

cos2 x

Separate the fraction:

=cos x

cos2 x+cos x sin x

cos2 x

Reduce the fractions:

=1

cos x+sin x

cos x

Next, simplify each fraction

sec x + tanxThe identity is now complete and so is the tutorial. See your teacher for practice problems.