how to simplify trigonometric identity problems

Master simplifying and proving trigonometric identities by working along with these 24 trig identity problems. I put the problems in the multiple-choice format so the problems are more challenging. Moreover, we will also derive the triple-angle identity, the quadruple-angle identity for cosine, i.e. cos(3x) and cos(4x), and the power reduction identity for sin^4(x). Mastering these problems will prepare you for your calculus class, especially when we do derivatives and integrals.
#trigonometry #math #precalculus
File 👉 / 24-trig-identity-88803595
Solutions 👉 / my-hand-written-88804184
🛍 Shop my math t-shirt & hoodies: amzn.to/3qBeuw6
Thanks to @takureido3122 & @sohampinemath1086 for the time stamps!!!
Intro - 00:00
1 - 00:51 - sinx+cotxcosx
2 - 04:40 - (secx-cosx)/sinx
3 - 07:16 - cotx/(cscx-sinx)
4 - 09:00 - (1+2cosx)/(2+secx)
5 - 10:45 - 1/(1-sinx)+1/(1+sinx)
6 - 13:00 - (2+cot²x)/csc²x-1
7 - 15:36 - tan(x+π/4)
8 - 19:11 - cos(3x)
9 - 24:09 - cos(4x)
10 - 29:32 - sec(sin^-1x)
11 - 32:42 - cos(2tan^-1x)
12 - 36:56 - tan(2sin^-1x)
13 - 42:44 - csc²x+sec²x
14 - 45:41 - cosx cos(2x)
15 - 49:38 - sin⁴x
16 - 57:22 - sin⁴x-cos⁴x
17 - 1:02:04 - tanx+tany
18 - 1:04:47 - 2tanx/(1+tan²x)
19 - 1:07:25 - 1/(secx-1)+1/(secx+1)
20 - 1:12:08 - (sinx+tanx)/(1+cos(-x))
21 - 1:16:14 - cos²x-sin⁴x sec²x
22 - 1:14:50 - (sinx+cosx)²
23 - 1:20:18 - sinx/(1-cotx)+cosx/(1+tanx)
24 - 1:23:57 - sec²(x/2)
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Thank you all!

[@sohampinemath1086]

Timestamps below 😊😊 1) 0:32 2) 4:42 3) 7:15 4) 9:04 5) 10:47 6) 13:01 7) 15:36 8) 19:10 9) 24:12 10) 29:33 11) 32:40 12) 36:55 13) 42:46 14) 45:41 15) 49:41 16) 57:25 17) 1:02:07 18 ) 1:04:49 19) 1:07:29 20) 1:12:09 21) 1:16:11 22) 1:14:50 23) 1:20:21 24) 1:23:58 100 differential equations please❤❤

[@blackpenredpen]

Thank you!!!!!!

[@demon-ow4vg]

@@blackpenredpen please make a video about Digamma (x)=1

[@imaginaryangle]

The amount of care you take that the viewer really gets it and doesn't trip over a step you are making is absolutely amazing

[@blackpenredpen]

Thank you!

[@blackpenredpen]

Could someone please help me do the time stamps because it will really help all the viewers to navigate the video? I will also credit your name in the description. Thank you.

[@shishirjha4219]

Ok then you have to wait for an hour and a half and I don't need any credits

[@thatomofolo452]

Chain effect 👍👍😁

[@L17_8]

Jesus loves you ❤️ please turn to him and repent before it's too late. The end times described in the Bible are already happening in the world.

[@shishirjha4219]

@@L17_8 bro is on drugs

[@thenew3dworldfan]

I have a question. Suppose a(n) is the integer sequence A007916. Does the series (1 to inf) 1/ (A007916(n))^2 conv or div?

[@ayssinaattori9313]

We just started doing trig equations in school and I realised how fun it is to play around with trig identities. Really glad I found this video, it was really fun and I became more familiar with with other trig functions than sin, cos, and tan, which are the ones we've manly had thus far.

[@ayssinaattori9313]

42:20 still watching, doing the questions and watching your explanation if I get stuck. Thank you for all your great videos!

[@JonnyMath]

Hi backpenredpen!!! You always keep me motivated to record math videos!!! Sometimes I ask myself why should I continue making math videos on youtube, but then I watch one of your videos and they inspire myself to do it!!!🤗🤗🤗

[@NoahMartin-ii1wz]

do 200 trig equations/inequalities in one take as fast as you can 😂😂

[@L17_8]

Jesus loves you ❤️ please turn to him and repent before it's too late. The end times described in the Bible are already happening in the world.

[@NoahMartin-ii1wz]

make it logarithmic equations/inequalities LOLL@@L17_8

[@user-fh2rw9zh8g]

I wish he doooesss I'm currently self studying trigonometry and this would be so helpful

[@robertcampomizzi7988]

Listen to the very last line of "Alphabet Aerobics" by Blackalicious and you'll know why I think you should.

[@robertcampomizzi7988]

​​@@L17_8if Jesus was a carpenter then he used trigonometry. Why didn't they write that trig down in the Bible? I'm just asking questions.

[@ahsonkusare2352]

Please next videos:24 indefinite integrals and 24 definite integrals (High level problems) And happy teachers day

[@r.i.p.volodya]

Many THANKS for all the time you put into your videos. I find you inspirational 😁

[@adnantasin485]

My interest in mathematics has grown so much because of you. You make so good content. Huge fan of your work. Thank you so much.

[@abdirahmansamow7451]

you are the goat, i was struggling with this topic then you popped into my recommended list and now im decent at it, thank you

[@blackpenredpen]

Glad to hear 😃

[@k3yb0ard73]

I love your videos, keep up the amazing content!!! Btw, I'm in Precalculus learning Trig Identities!

[@user-fh2rw9zh8g]

Hello!! I've just finished watching 50 integrals video, and I'm so happy to see this as I'm struggling with trigonometry!

[@hawtpotato90210]

42:40 I am with you brother. I am doing my best on the problems and you're filling in the gaps as I work on this over the course of a few days. Your videos have been valuable to my efforts to strengthen my math skills. Thank you very very very much BpRp!

[@blackpenredpen]

Glad to hear!! Keep up your good work and best wishes to you!

[@franciscomorales8102]

My Lord!!! Thank you very much!!! You had simplified my life!! I’m back in school after many years, and I was struggling into I found your video!!! Where did you got the T-shirt and the table you have on the wall??? I appreciate your time, dedication and advice!!!

[@professionalgster1176]

Thank you so much for explaining well, I was assigned this today and was so confused and SCARED haha. Watched over 2 of your problems and I'm starting to understand!

[@ayssinaattori9313]

Finished the video now, took some breaks in between. Was able to get most questions on my own but it was really fun to see how our ways of solving them differed. Especially on questions 16 and 17 I overcomplicated it a lot but I did end up at the right answer. Tank you so much!

[@blackpenredpen]

Glad to hear. Thank you, too!

[@ITWorksSoftware]

42:40 yes, doing it with you. I've also done the 100 derivatives twice now, and the 100 algebra twice, too... Great practice!

[@jaii5955]

I definitely loved your math questions ❤😮

[@anurakumara3950]

Thanks BpRp you're one of the great maths teacher 😎🖤❤

[@kirbo722]

This video was VERY helpful, thank you!!

[@RishavPlayz2007]

42:40 yes i am following along solving all the questions, they're pretty easy, excited for the upcoming questions

[@user-pc1zf9gh3o]

Hey man you are really an inspiration for me.your passion keeps me motivated ❤

[@cooldown6077]

Thank you so much. This will help a lot in my calculus studies. 🙂

[@blackpenredpen]

Glad to hear that!

[@dannyfarias6607]

Thank you so much for this amazing video

[@tariqallali4507]

Thanks prof i really learned a bunch of things

[@frozenpeak1524]

I'm taking AP Precalculus this year and so far its not bad. So far its just been obtaining a deeper understanding of functions and what values can be obtained from the equation of a function.

[@kylewaddle7592]

I dont comment often but i felt that i should mention how helpful and entertaining with me learning calculus and other math. Also i loved this video

[@blackpenredpen]

Thank you!

[@Dmoney_411]

​@@blackpenredpenHey I am taking AP precalculus, we are over hallways through unit 2 and it doesn't seem too bad. any suggestions on how to master my algebra (especially algebra 2) skills that are lacking from having those classes over zoom. Thank you 🙏

[@seshansahoo7913]

Happy Teacher's day Sir. Today 5th September every year in India we celebrate Teacher's Day to so our love toward the teachers. Your lecture are just brilliant i following your channel since 2020.thak you sir for everything.

[@scottleung9587]

Awesome job!

[@saurabpoudel9134]

Thanks professor ! Please also make video on analytical geometry

[@mikeheyburn9716]

This is terrific Steve

[@marshallmanz123]

You deserve more views man.

[@demiryumruk1423]

for q6 an alternative solution would be to take the -1 as being equal to -sin^2(x)-cos^2(x) (multiplying original identity by -1) and substituting, then simplifying the equation down to sin^2(x) really helpful video, i love all your tutorials

[@foolwhodreams]

thank u for this. i took ap calculus bc and skipped calc 1 and 2, and im scared of trig identities coming up in chapter 12 for calculus 3 !! big fan

[@user-pb5kr4qe9q]

Master of it ❤

[@ahmet00fx]

i am a 12th grade student in Turkey and these questions are way easy than I expected.

[@dinoortolani6794]

you're amazing thanks

[@crazyteamoffici7556]

Love you sir ❤❤❤❤

[@danielputrahandana2982]

I watch these while eating because I dont know what Video to watch

[@rubensf7780]

1:03:31 Easy, cancel cos from both sides

[@a.guzman1512]

What is that reference board you are using and where can I get it? Great video!

[@Subham-Kun]

49:24 "If I did my math right" - If this man makes mistakes in addition, I am allergic to numbers

[@datarioplays]

Do more integral battles between elementary and non-elementary ones! The one you did 4 years ago wasn’t enough for me.

[@henridelagardere264]

24 Trig Problems done in *one* take, that's gotta be BPRP, aka 'The Beast'. Identity mathematics presented the way we spoilt brats have grown used to. Don't take this man for granted, though!

[@L17_8]

Jesus loves you ❤️ please turn to him and repent before it's too late. The end times described in the Bible are already happening in the world.

[@themultiroundergamer6073]

hey whats that black identity board you have hanging beside the whiteboard? is it available online?

[@yoshikagekiraaaaa]

nice!

[@ChiefToad619]

I lose sleep over these videos, even though I may or may not know the content😅

[@thenew3dworldfan]

I have a question. Suppose a(n) is the integer sequence A007916. Does the series (1 to inf) 1/ (A007916(n))^2 conv or div?

[@ArthurvanH0udt]

I would love it if you’d do some extra identities for the parabolic version of trig fies. So cosp and sinp identities! Why? Cause they behave somewhat different than the normal sin/cos and hyperbolic sinh/cosh versions!

[@samarjitdas6378]

Happy Teacher's Day Love from India ❤ 🇮🇳

[@mitthrawnuruodo2880]

Happy teacher's day sir🎉

[@prajjwalnarayan8001]

Amazing

[@yugam6578]

Could you please make a video on jee advanced maths section

[@aali-fg8mg]

I love you man

[@takureido3122]

Intro - 00:00 1 - 00:51 - sinx+cotxcosx 2 - 04:40 - (secx-cosx)/sinx 3 - 07:16 - cotx/(cscx-sinx) 4 - 09:00 - (1+2cosx)/(2+secx) 5 - 10:45 - 1/(1-sinx)+1/(1+sinx) 6 - 13:00 - (2+cot²x)/csc²x-1 7 - 15:36 - tan(x+π/4) 8 - 19:11 - cos(3x) 9 - 24:09 - cos(4x) 10 - 29:32 - sec(sin^-1x) 11 - 32:42 - cos(2tan^-1x) 12 - 36:56 - tan(2sin^-1x) 13 - 42:44 - csc²x+sec²x 14 - 45:41 - cosx cos(2x) 15 - 49:38 - sin⁴x 16 - 57:22 - sin⁴x-cos⁴x 17 - 1:02:04 - tanx+tany 18 - 1:04:47 - 2tanx/(1+tan²x) 19 - 1:07:25 - 1/(secx-1)+1/(secx+1) 20 - 1:12:08 - (sinx+tanx)/(1+cos(-x)) 21 - 1:16:14 - cos²x-sin⁴x sec²x 22 - 1:14:50 - (sinx+cosx)² 23 - 1:20:18 - sinx/(1-cotx)+cosx/(1+tanx) 24 - 1:23:57 - sec²(x/2)

[@blackpenredpen]

Thank you so much!

[@talentlogic1512]

I like this a lot. Thank you my friend.

[@sajuvasu]

Love you sir....❤️❤️❤️ From india We aren't studying Trigonometry.. But i understood all you taught.... On the 14th question i think we can simplify it to Cos(x)Cos(2x)=2Cos³(x)-Cos(x) Its heartbreaking to see the expression being not factorised... 🤣🤣 Any way.. Big fan sir You are wonderful...🔥🔥❤️❤️ (And Happy teacher's day also❤️)

[@ManasvinShrimali]

at 14:37 sir you can also write 2sin^2(x) as sin^2(x) + sin^2(x) + cos^2(x) - 1=sin^2(x)

[@TheZerovirus1000]

42:39 I could watch a movie of these

[@blackpenredpen]

Thanks!!!

[@notboboi9977]

1:06 if u look at the captions.. cosx over se-

[@illumexhisoka6181]

7 can be simplified to tan(2x)+sec(2x) and 13 can be 4sec^2(2x)

[@suryatejacv3042]

Hai sir can you discuss telescopic sums with alternating positive negative sums

[@zoro8800]

these were very easy as I am in high school

[@TanmaY_Integrates]

Do a 100 question One Shot version and put in 100 question playlist

[@jitenderbajaj1808]

Plz solve Integral of log base 2(x*(sin2ex))

[@antonioaccogli1014]

I am doing to do your test to repeat for my school test... i will tell you how it will go

[@ChristianRosenhagen]

13 is stunning

[@lgooch]

Do inverse trig identities next!

[@stavzee]

His smile just makes me wanna study more

[@brucefrizzell4221]

I pick up a lot of small details in these longer videos!

[@blackpenredpen]

Glad to help!

[@carterbagvitch5964]

I ran into an interesting trig problem on my no calculator pre-calc/alg 2 test and was wondering how to do it since I only had solutions and all my teachers said there had to be a trick they didn’t know. The problem was If sin(x)+cos(x)=1/3, what does ((sin(x))^5)+((cos(x))^5)=

[@adamhurt6140]

what was the answer they gave you?

[@carterbagvitch5964]

@@adamhurt6140 I figured it out. The answer was 344, but thats cause they asked for the answer in fractional form (k/w), and then the answer is k+w. thats obviously not a critical step, so I didnt include it. If you want to know, u take pascals triangle and expand it out with sin(x) and cos(x), isolate sin^5(x) +cos^5(x), then factor and break down the rest of it using trig identities and the given. Also have to realize that since sin(x)+cos(x)=1/3, cos(x)*sin(x)=-4/9. thats also key

[@walter9029]

Is ex. 2 an identity? Because in the original fraction all values equal to kxpi are not in the domain.

[@salvadorsabateromero2258]

18:55

[@flazeanimations]

Idk why I always feel satisfied when he erased the whiteboard 😂

[@divinitas604]

Where do I find the identities board?

[@vivekgupT______________]

Bro you are awesome I love you ❤❤ I am from india and I am like maths

[@blackpenredpen]

Thank you!

[@vivekgupT______________]

Bro can help me a little bit

[@Subham-Kun]

26:20 It is unbelievable that even the GOAT of Mathematics, who alone is strong enough to beat Lord Wolfram|Alpha, is lagging!!!!!!!!!!! THIS IS ALARMING FOR THE DIMENSION OF MATHEMATICS!!!!!!!!!!!!

[@_maxmath]

Hi bprp can u please do a video to show us how to calculate cos(x+y+z)?

[@Apollorion]

First you add x, y and z together, then you calculate the cosine of that sum. Or is that not what you mean? edit: e.g. do you want a formula that distributes the x, y and z into their 'own' trigonometric functions? If so then just apply the sum formula's for the cosine and sine i.e. sin(a+b)=sin(a)cos(b)+sin(b)cos(a) cos(a+b)=cos(a)cos(b)-sin(a)sin(b) and of course, begin with splitting one of the x,y and z terms off from the other two, and after that split the sine and cosine that other pair (and that's why I also mentioned the sum formula for the sine).

[@roof310]

can you do inequalities please

[@SmSitutul]

broo ❤❤

[@FVerz]

26:34 Bro acts like he just made the height of 2 tall buildings approach 0m in 2001

[@sapoman01]

my comment down below at the 42:39 time stamp ⏯

[@DazHuang72]

Calc 2 savior

[@PIKI146]

I now prefer to stay home and do calculus on my weekends

[@nemanjalazarevic9249]

My take: 1. sinx +cotxcosx = sinx +(cos^2x/sinx) = (sin^2x + cos^2x/sinx) = 1/sinx = cscx 2. (1-cos^2x)/sinxcosx = tanx 3.cosx/sinx(1/sinx - sinx) = cosx/1-sin^2x = secx 4:1+2cosx/2+secx = 1+2cosx/2+1/cosx = cosx+2cos^2x/2cosx+1 =cosx(2cosx+1)/2cosx+1 = cosx 5:1+sinx/(1-sinx)(1+sinx) + 1-sinx/(1-sinx)(1+sinx) = 2/cos^2x = 2sec^2x

[@user-kd7to5xf9e]

Ok time to do some maths

[@amarendranathhazra4142]

Do all the 24 identities you did but hyperbolic trig instead

[@lordpz3518]

plz teach mathematicalinduction

[@gamerpedia1535]

Cool thing to note: 8cos⁴(x) - 8cos²(x) + 1 = 8sin⁴(x) - 8sin²(x) + 1

[@Tomuchfanforfon]

Good teacher watching from india guys reply where are you watching from?

[@carultch]

I like your Nepal flag.

[@Bambambami]

This is driving me crazy i wanna cry so bad

[@joaquinnunez7272]

you can do it my friend

[@user-fh2rw9zh8g]

I didn't skip a whole 10 min ad 😂😂 hope this counts for you, because i don't have money to send

[@sohampinemath1086]

Please do 100 differential equations I need them for my exams😭😭

[@amitlanis3104]

yesss that would be super helpful😁

[@L17_8]

Jesus loves you ❤️ please turn to him and repent before it's too late. The end times described in the Bible are already happening in the world.

[@parv840]

12 th m h kya bhai

[@Sanatan_saarthi_1729]

Challenge:- solve 100 ordinary differential equations in one go,,,,next 100 partial differential equations in one go

[@wearealiens2050]

😂Imagine, sir, that you unplug the camera lol

[@n0nebtw]

art

[@nvapisces7011]

7:30 lemme just take a look at the camera (nope he is definitely taking a look at the solution jk)

[@wearealiens2050]

I have an equation who solves it ‏‪13:19‬‏

[@davidthomas6043]

I don't understand why he puts cos3x = (cosx)(cos2x)..This seems wrong, because for example, cos 60 = o.5, but cos 20 is .9397 and cos 40 is .766, and multiplying these these gives an answer of of .7198. which is nowhere near .5 ! Can someone please explain? I Thanks

[@AttyPatty3]

Hey David! He didn't put cos 3x = (cosx) (cos2x) instead what he did was simply write the 3x as 2x + x. cos(3x) = cos(2x + x)

[@davidthomas6043]

Thanks for answering. I now see that what he wrote was (cos x)(cos2x- minus (sin x)(sin 2x). Silly me !! @@AttyPatty3

[@davidthomas6043]

Further to mt previous reply, can you please tell me if there is any way available to derive the formula cos (A + B) = cos A.cos B - sin A.sin B ?@@AttyPatty3