Introduction:

Trigonometric Identities are equations that involve trigonometric functions (sin(x), cos(x), tan(x), csc(x), sec(x), and cot(x) and are true for every single value of the variables where both sides of the equal sign are defined. These are equations that are always true involving certain functions of one or more angles. An example of an identity is the Pythagorean theorem a^{2}+b^{2}=c^{2}

- Example:

cscx * tanx = secx

Rules for Verifying Identities

No algebra across the =

Work is done on one side (the more complicated side)

Verifying is not done until both sides are exactly the same

Make sure you show all of your work!

You should remember how to find identities, rather than memorize each one.

**Reciprocal ID’s:**

Each trigonometric function has a reciprocal of itself. A reciprocal of a number is 1 divided by that number

$csc(x)=1/sin(x)$

The reciprocal of sine is cosecant, this ID shows what cosecant is in terms of sine.

$sec(x)=1/cos(x)$

The reciprocal of cosine is secant, this ID shows what secant is in terms of cosine.

$cot(x)=1/tan(x)$

The reciprocal of tangent is cotangent, this ID shows what cotangent is in terms of tangent.

**Quotient ID's:**

Consider the following diagram where the point (*x, y*) defines an angle θ at the origin, and the distance from the origin to the point is r units:

From the diagram, we can see that the ratios sin θ and cos θ are defined as:

$sin(θ)=y$

And

$cos(θ)=x/r$

Now, we use these results to find a definition for tan θ:

$cos(θ)*sin(θ)=(y/r)/(r/x) = y/r*r/x=y/x$

Now also $tan(θ)= y/x$ so we can conclude:

$tan(x)=sin(x)/cos(x)$

$cot(x)=cos(x)/sin(x)$

- Example:

Verify sin(x)*sec(x)=tan(x)

- $(sin(x)/1)*(1/cos(x))$ Use the corresponding trig ID's for sin(x) and sec(x).
- $sin(x)/cos(x)$ Multiply the IDs together.
- $tan(x)=tan(x)$ The corresponding trig ID for $sin(x)/cos(x)$ is tan(x).

Practice Problems. Verify Each

- $cosx(tan(x) + cot(x)) = csc(x)$
- $1-sin(x)/cos(x)+ cos(x)/1-sin(x)= 2sec(x)$
- $tan(x)-cot(x)/tan(x)+cot(x)= sin^2(x)+sin(x)$

**Pythagorean ID's:**

Looking back at the diagram above, using the Pythagorean theorem, we can conclude that:

$y^2+x^2=r^2$

dividing through by r^2 would equal:

$y^2/r^2+x^2/r^2=1$

This can then equal:

$1=sin^2(x)+cos^2(x)$

By dividing this equation by cos^2(x), we get:

$sin^2(x)/cos^2(x)=1/cos^2(x)$

This would then equal:

$cos(2x)=1-sin(2x)$

And finally, by dividing by sin^2(x), we get:

$sin^2(x)=1-cos^2(x)$

Other pythagorean identities are found the same way:

$csc^2(x)=cot^2(x)+1$

$cot^2(x)=csc^2(x)-1$

$1=csc^2(x)-cot^2(x)$

$tan^2(x)=sec^2(x)-1$

$sec^2(x)=tan^2(x)+1$

$1=sec^2(x)-tan^2(x)$

**Even and Odd ID’s**

$sin(-x)=-sin(x)$

$cos(-x)=cos(x)$

$tan(-x)=tan(x)$

$csc(-x)=-csc(x)$

$sec(-x)=sec(x)$

$cot(-x)=-cot(x)$

**Sum and Difference ID’s**

$cos(x+y)=cos(x)cos(y)-sin(x)sin(y)$

$cos(x-y)=coscos+sinsin$

$sin(x+y)=sin(x)cos(y)+cos(x)sin(y)$

$sin(x-y)=sin(x)cos(y)-cos(x)sin(y)$

$tan(x+y)=tan(x)+tan(y)/1-tan(x)tan(y)$

$tan(x-y)=tan(x)-tan(y)/1+tan(x)tan(y)$

Practice Problems: Verify Each

- $sin(x+y) = cos((x+b)-π/2)$
- $tan(x+y) =sin(x-y)/cos(x+b)$
- $cos(x-y)/sin(x)sin(y)= cot(x)cot(y)+1$

**Double Angles ID’s**

$sin(2x)=2sin(x)cos(x)$

$cos(2x)=cos^2(x)-sin^2(x)$

$cos(2x)=1-2sin^2(x)$

$cos(2x)=2cos^2(x)-1$

$tan(2x)=2tan(x)/1-tan^2(x)$

**Half-Angle IDs**

$sin(x/2)= +/- root(1-cos(x)/2)$

$cos(x/2)= +/- root(1+cos(x)/2)$

$tan(x/2)= +/- root(1-cos(x)/1+cos(x))$

**Product to Sum IDs**

$cos(x)sin(y)=1/2{win(x-y)-sin(x+y)}$

$cos(x)cos(y)=1/2{cos(x+y)+cos(x-y)}$

$sin(x)cos(y)=1/2{sin(x+y)+sin(x-y)}$

$sin(x)sin(y)=1/3{cos(x-y)-cos(x+y)}$

**Sum to Product IDs**

$sin(x)+sin(y)=2sin(x+y/2)cos(x-y/2)$

$sin(x)-sin(y)=2sin(x-y/2)cos(x+y/2)$

$cos(x)+cos(y)=2cos(x+y/2)cos(x-y/2)$

$cos(x)-cos(y)=-2cos(x+y/2)sin(x-y/2)$