We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Trigonometry : Trigonometric Identities

Study concepts, example questions & explanations for Trigonometry

Create An Account

All Trigonometry Resources

6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #11 : Trigonometry

Reduce the following expression.

$\small \frac{\sin ^{2} x + \cos^{2} x}{\sin x}$

Possible Answers:

$\csc x$

$\sec x$

$1+\cot ^{2} x$

Correct answer:

$\csc x$

Explanation:

Because $\sin ^{2} x + \cos ^{2} x = 1$ ,

therefore:

$\small \frac{\sin ^2 x + \cos^2 x}{\sin x} = \frac{1}{\sin x}$

By definition of cosecant,

$\small \frac{1}{\sin x} = \csc x$

Report an Error

Example Question #12 : Pythagorean Identities

Reduce the following expression.

$\small \tan ^2 x +1 - \sec^2 x$

Possible Answers:

$\small -\sec x$

$\small 2\tan ^2 x$

$\small \sec ^2 x - 1$

Correct answer:

Explanation:

There are several ways to work this problem, but all of them use the second Pythagorean trig identity, $\small \tan ^2 x + 1 = \sec^2 x$ .

You can use this identity to substitute for parts of the expression. Here are two examples.

Method 1:

Substituting $-(\tan^2 x +1)$ for $\small -\sec^2 x$ , we get

$\tan ^2 x +1 - (\tan ^2 x +1)$ , which equals zero.

Method 2:

Substituting $\sec^2 x$ for $\tan^2 x +1$ , we get

$\sec^2 x-\sec^2 x$ , which equals zero.

Regardless of which substitution you choose, the answer is the same.

Report an Error

Example Question #11 : Pythagorean Identities

Simplify

$\frac{\sin^{3}{\left (x \right )} + \cos{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

Possible Answers:

$\frac{1}{2} \cos{\left (x \right )} \csc^{2}{\left (x \right )}$

$\cos^{2}{\left (x \right )} + \csc^{3}{\left (x \right )}$

$\frac{1}{2} \sin{\left (x \right )}$

$\frac{1}{2} \left(\sin{\left (x \right )} + \cos{\left (x \right )} \csc^{2}{\left (x \right )}\right)$

$\sin^{2}{\left (x \right )} + \sec^{2}{\left (x \right )}$

Correct answer:

$\frac{1}{2} \left(\sin{\left (x \right )} + \cos{\left (x \right )} \csc^{2}{\left (x \right )}\right)$

Explanation:

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{3}{\left (x \right )} + \cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} \sin{\left (x \right )} + \frac{\cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

$\\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}$

Now simplify, and combine each term.

$\\\frac{1}{2} \sin{\left (x \right )} + \frac{1}{2} \cos{\left (x \right )} \csc^{2}{\left (x \right )} \\\\\frac{1}{2} \left(\sin{\left (x \right )} + \cos{\left (x \right )} \csc^{2}{\left (x \right )}\right)$

Report an Error

Example Question #11 : Trigonometry

Simplify

$\frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

Possible Answers:

$\frac{1}{2}$

$\sin^{2}{\left (x \right )} + \sec^{2}{\left (x \right )}$

$\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

$\cos^{2}{\left (x \right )} + \csc^{3}{\left (x \right )}$

$\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )}$

Correct answer:

$\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Explanation:

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} + \frac{\cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

Now simplify, and combine each term.

$\\\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\\\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Report an Error

Example Question #13 : Trigonometry

Simplify

$\frac{\sin^{2}{\left (x \right )} + \cos{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

Possible Answers:

$\frac{1}{2}$

$\sin^{2}{\left (x \right )} + \sec^{2}{\left (x \right )}$

$\cos^{2}{\left (x \right )} + \csc^{3}{\left (x \right )}$

$\frac{1}{2} \cos{\left (x \right )} \csc^{2}{\left (x \right )}$

$\frac{1}{2} \left(\cos{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Correct answer:

$\frac{1}{2} \left(\cos{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Explanation:

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{2}{\left (x \right )} + \cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} + \frac{\cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

Now simplify, and combine each term.

$\\\frac{1}{2} \cos{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\\\frac{1}{2} \left(\cos{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Report an Error

Example Question #12 : Trigonometry

Simplify

$\frac{\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

Possible Answers:

$\cos^{2}{\left (x \right )} + \csc^{3}{\left (x \right )}$

$\frac{1}{2} \left(\cos^{2}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

$\frac{1}{2}$

$\sin^{2}{\left (x \right )} + \sec^{2}{\left (x \right )}$

$\frac{1}{2} \cos^{2}{\left (x \right )} \csc^{2}{\left (x \right )}$

Correct answer:

$\frac{1}{2} \left(\cos^{2}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Explanation:

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} + \frac{\cos^{2}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

Now simplify, and combine each term.

$\\\frac{1}{2} \cos^{2}{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\\\frac{1}{2} \left(\cos^{2}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Report an Error

Example Question #17 : Pythagorean Identities

Simplify

$\frac{\sin^{3}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

Possible Answers:

$\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )}$

$\cos^{2}{\left (x \right )} + \csc^{3}{\left (x \right )}$

$\sin^{2}{\left (x \right )} + \sec^{2}{\left (x \right )}$

$\frac{1}{2} \sin{\left (x \right )}$

$\frac{1}{2} \left(\sin{\left (x \right )} + \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )}\right)$

Correct answer:

$\frac{1}{2} \left(\sin{\left (x \right )} + \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )}\right)$

Explanation:

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{3}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} \sin{\left (x \right )} + \frac{\cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

Now simplify, and combine each term.

$\\\frac{1}{2} \sin{\left (x \right )} + \frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} \\\\\frac{1}{2} \left(\sin{\left (x \right )} + \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )}\right)$

Report an Error

Example Question #18 : Pythagorean Identities

Simplify

$\frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

Possible Answers:

$\cos^{2}{\left (x \right )} + \csc^{3}{\left (x \right )}$

$\frac{1}{2}$

$\sin^{2}{\left (x \right )} + \sec^{2}{\left (x \right )}$

$\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

$\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )}$

Correct answer:

$\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Explanation:

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} + \frac{\cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

Now simplify, and combine each term.

$\\\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\\\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Report an Error

Example Question #19 : Pythagorean Identities

Simplify

$\frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

Possible Answers:

$\cos^{2}{\left (x \right )} + \csc^{3}{\left (x \right )}$

$\frac{1}{2}$

$\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

$\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )}$

$\sin^{2}{\left (x \right )} + \sec^{2}{\left (x \right )}$

Correct answer:

$\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Explanation:

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} + \frac{\cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

Now simplify, and combine each term.

$\\\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\\\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Report an Error

Example Question #20 : Pythagorean Identities

Simplify

$\frac{\sin^{2}{\left (x \right )} + \cos{\left (x \right )}}{2 \cos^{2}{\left (x \right )} - 1}$

Possible Answers:

$\frac{1}{2}$

$\frac{1}{2} \left(\cos{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

$\cos^{2}{\left (x \right )} + \csc^{3}{\left (x \right )}$

$\frac{1}{2} \cos{\left (x \right )} \csc^{2}{\left (x \right )}$

$\sin^{2}{\left (x \right )} + \sec^{2}{\left (x \right )}$

Correct answer:

$\frac{1}{2} \left(\cos{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Explanation:

The first step to simplifying is to remember an important trig identity.

$\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1$

If we rewrite it to look like the denominator, it is.

$\sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1$

Now we can substitute this in the denominator.

$\frac{\sin^{2}{\left (x \right )} + \cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Now write each term separately.

$\frac{1}{2} + \frac{\cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}$

Remember the following identities.

$\\\frac{1}{2} \cos{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\\\frac{1}{2} \left(\cos{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)$

Report an Error

View Trigonometry Tutors

Dalton
Certified Tutor

University of Massachusetts Amherst, Doctor of Philosophy, Mathematics.

View Trigonometry Tutors

Vandana
Certified Tutor

Jawaharlal Nehru Technological University, Bachelor of Technology, Civil Engineering. Florida International University, Maste...

View Trigonometry Tutors

Vicky
Certified Tutor

Clemson University, Doctor of Science, Civil Engineering.

All Trigonometry Resources

6 Diagnostic Tests 155 Practice Tests Question of the Day Flashcards Learn by Concept

Trigonometry Tutors in Top Cities:

Popular Courses & Classes

SAT Courses & Classes in Denver, LSAT Courses & Classes in Seattle, SAT Courses & Classes in Phoenix, ACT Courses & Classes in Los Angeles, ACT Courses & Classes in Philadelphia, ACT Courses & Classes in Houston, MCAT Courses & Classes in Washington DC, GMAT Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Phoenix, GRE Courses & Classes in Los Angeles

Popular Test Prep

LSAT Test Prep in Boston, ACT Test Prep in Philadelphia, ACT Test Prep in Houston, GRE Test Prep in Philadelphia, ACT Test Prep in Los Angeles, SAT Test Prep in Dallas Fort Worth, MCAT Test Prep in Miami, LSAT Test Prep in San Diego, ACT Test Prep in Chicago, GMAT Test Prep in New York City

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Trigonometry : Trigonometric Identities

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #11 : Trigonometry

Example Question #12 : Pythagorean Identities

Example Question #11 : Pythagorean Identities

Example Question #11 : Trigonometry

Example Question #13 : Trigonometry

Example Question #12 : Trigonometry

Example Question #17 : Pythagorean Identities

Example Question #18 : Pythagorean Identities

Example Question #19 : Pythagorean Identities

Example Question #20 : Pythagorean Identities

All Trigonometry Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: