Trig Identity

What are Trig Identities?

Reciprocal Identities: 

Quotient Identities:

Multiple Select

Which statements are true?

• $$\sin x=\frac{1}{\cos x}$$

• $$\tan^2x=\frac{\sin^2x}{\cos x}$$

• $$\sec x=\frac{1}{\cos x}$$

• $$\cot x=\frac{\cos x}{\sin x}$$

• $$\csc^2x=\frac{1}{\sin^2x}$$

Pythagorean identity #1:

Pythagorean Identity #2

Pythagorean Identity #3

Anymore Pythagorean Identities??

Yes! Even more pythagorean identities can be obtained by rearranging (adding/subtracting) within the 3 identities we just covered!

See if you can figure out what they are from the following questions....

Multiple Choice

Rearrange:  $$\sin^2x+\cos^2x=1$$  to solve for:   $$\sin^2x$$   

• $$\sin^2x=1-\cos^2x$$  

• $$\sin^2x=1+\cos^2x$$  

• $$\sin^2x=\cos^2x-1$$  

Multiple Choice

Rearrange:  $$\sin^2x+\cos^2x=1$$  to solve for  $$\cos^2x$$  

• $$\cos^2x=1-\sin^2x$$  

• $$\cos^2x=1+\sin^2x$$  

• $$\cos^2x=\sin^2x-1$$  

Multiple Choice

Rearrange to solve for $$\tan^2\theta$$ : 
                                        $$1\ +\ \tan^2\theta=\ \sec^2\theta$$  

• tan²θ = sec²θ + 1

• tan²θ = sec²θ - 1

• tanθ = secθ - 1

• tan²θ = opp/adj

Multiple Choice

Rearrange so it equals 1: 
                                        $$1\ +\ \tan^2x=\ \sec^2x$$  

• $$\tan^2x-\sec^2x=1$$  

• $$\sec^2x-\tan^2x=1$$  

• $$\sec^2x+\tan^2x=1$$  

• $$\tan^2x-1=\sec^2x$$  
  

Multiple Choice

Solve the following for  $$\cot^2\theta$$  :
$$1+\cot^2\theta=\csc^2\theta$$  

• 1 = cot²θ + csc²θ

• cot²θ =1 - csc²θ

• cot²θ = csc²θ -1

• cot²θ = adj/opp

Multiple Choice

Rearrange so it equals 1 :
$$1+\cot^2\theta=\csc^2\theta$$  

• $$1=\cot^2\theta+\csc^2\theta$$

• $$\frac{\cot^2\theta}{\csc^2\theta}=1$$  

• $$\cot^2\theta-\csc^2\theta=1$$  
  

• $$\csc^2\theta-\cot^2\theta=1$$  

IN CASE YOU DIDN'T GET THEM....HERE ARE ALL THE PYTHAGOREAN ARRANGEMENTS!

How do we use these identities?

Multiple Choice

Exchange each term with an identity then use algebra to reduce:

                                 $$\frac{\tan x}{\sec x}$$

• $$\cos x$$  

• $$\csc x$$  

• $$\cot x$$  

• $$\sin x\$$  

Multiple Choice

Now try this one.  Again reduce by substituting with identities:
$$\left(\sec x\right)\left(\cot x\right)\left(\sin x\right)$$

• sin x

• - sin x

• 1

• -1

Simplifying Suggestions

• Here are some common techniques for simplifying trig expressions

• Notice you will be using algebra to help reduce

• Let's look at more problems using these suggestions......

Multiple Choice

Reduce by substituting with identities:

$$\tan\ x\ \cot x\ -\cos^2x$$  

• tanx

• cotx

• sin²x

• cos²x

Now try some on your own using the hints provided....

​

Multiple Choice

Simplify by factoring out a GCF first. Look at what you have left.  Can you replace with an identity?

$$\cos^3x+\sin^2x\cos x$$

• $$\sin x$$  

• $$\cos x$$  

• $$\tan x$$  

• 1

Multiple Choice

Distribute then exchange with identities:
$$\sin x\left(\sec x-\csc x\right)$$

• $$1-\tan x$$  

• $$\tan x-1$$  

• $$1-\cot x$$  

• $$\sin x$$  

Multiple Choice

Simplify by first splitting into 2 fractions: (Note:  you CANNOT cross out anything until you've done this!)

$$\frac{\cos x-\sin x}{\sin x\cos x}$$  

• $$\csc x-\sec x$$  

• 1

• 0

• $$\sec x-\csc x$$  

Multiple Choice

Substitute then find a common denominator so you can combine into 1 fraction:              $$\cos x+\sin x\tan x$$  

• csc x

• sec x

• 1/sec x

• cos x

Multiple Choice

Exchange each with identities, then get a common denom. on the bottom.  Once you have a complex fraction, flip and multiply.

    $$\frac{\csc x\cos x}{\tan x+\cot x}$$  

• cos²x

• sin²x

• cot²x

• tan²x

Now you know the basics!

• Tips for success:

• Memorize the identities so you can recall them quickly!

• Don't give up! There may be more than one way to simplify.

• Keep practicing! You got this!