Soal Identitas Trigonometri Tersulit di Dunia Beserta Penyelesaiannya

Mungkin layak soal ini mendapatkan ‘gelar’ tersebut. Soal ini saya peroleh dari Yahoo Answers Amerika Serikat setahun yang lalu dari seorang dosen matematika. Sayangnya saya lupa tautan (link) nya. Saya bilang luar biasa sang pembuat soal ini! Soal tersebut adalah (saya modifikasi terjemahannya dalam bahasa Indonesia) karena saya lupa bunyinya, Buktikan bahwa:

\displaystyle \frac{4\cos^4 x-4\sin x\cos^3 x-2\sin^2 x\tan x-\tan^2 x-2\cos 2x}{\sec^2x+\cos 2x}=\cos 2x-\sin 2x

 

Jawab:

\displaystyle \frac{4\cos^4 x-4\sin x\cos^3 x-2\sin^2 x\tan x-\tan^2 x-2\cos 2x}{\sec^2x+\cos 2x}\\= \frac{2\cos^2 x}{2\cos^2 x}~\frac{4\cos^4 x-4\sin x\cos^3 x-2\sin^2 x\frac{\sin x}{\cos x}-\frac{\sin^2 x}{\cos^2 x}-2\cos 2x}{\frac{1}{\cos^2 x}+\cos 2x}\\ =\frac{8\cos^6 x-8\sin x\cos^5 x-4\sin^3 x\cos x-2\sin^2 x-4\cos 2x\cos^2 x}{2+2\cos^2 x\cos 2x}\\ =\frac{(2\cos^2 x)^3-(2\sin x\cos x)(2\cos^2 x)^2-(2\sin x\cos x)(2\sin^2 x)-2\sin^2 x-(2\cos 2x)(2\cos^2 x)}{2+(\cos 2x)(2\cos^2 x)}\\ =\frac{(1+\cos 2x)^3-(\sin 2x)(1+\cos 2x)^2-(\sin 2x)(1-\cos 2x)-(1-\cos 2x)-(2\cos 2x)(1+\cos 2x)}{2+\cos 2x(1+\cos 2x)}\\=\frac{(1+3\cos 2x+3\cos^2 x+\cos^3 2x)-(\sin 2x)(1+2\cos 2x+\cos^2 x)-(\sin 2x)(1-\cos 2x)-(1-\cos 2x)-(2\cos 2x)(1+\cos 2x)}{2+\cos 2x+\cos^2 x}\\ =\frac{1+3\cos 2x+3\cos^2 x+\cos^3 2x-\sin 2x-2\sin 2x\cos 2x-\sin 2x\cos^2 x-\sin 2x+\sin 2x\cos 2x-1+\cos 2x-2\cos 2x-2\cos^2 2x}{2+\cos 2x+\cos^2 x}\\ =\frac{2\cos 2x+\cos^2 2x+\cos^3 2x-2\sin 2x-\sin 2x\cos 2x-\sin 2x\cos^2 2x}{2+\cos 2x+\cos^2 x}\\ =\frac{\cos 2x(2+\cos 2x+\cos^2 2x)-\sin 2x(2+\cos 2x+\cos^2 2x)}{2+\cos 2x+\cos^2 x}\\ =\frac{(\cos 2x-\sin 2x)(2+\cos 2x+\cos^2 2x)}{(2+\cos 2x+\cos^2 2x)}\\ =\cos 2x-\sin 2x
Terbukti!

Iklan

39 pemikiran pada “Soal Identitas Trigonometri Tersulit di Dunia Beserta Penyelesaiannya

  1. Undeniably believe that which you stated.
    Your favourite justification appeared to be at the net the easiest factor
    to be aware of. I say to you, I certainly get irked whilst folks consider issues that they plainly don’t know about.
    You controlled to hit the nail upon the top as neatly as defined out the whole thing
    with no need side-effects , other people can take a signal.
    Will probably be back to get more. Thank you

    Suka

  2. What i do not realize is in fact how you are not actually much more smartly-appreciated than you may be right
    now. You’re so intelligent. You already know thus significantly
    in the case of this subject, made me personally believe it from so many varied angles.
    Its like women and men aren’t fascinated except it’s one thing to
    accomplish with Lady gaga! Your personal stuffs excellent.
    All the time maintain it up!

    Suka

  3. My brother suggested I might like this blog.

    He was entirely right. This post truly made my day. You
    can not imagine simply how much time I had spent for this information!
    Thanks!

    Suka

  4. It’s very easy to find out any matter on net as compared to books,
    as I found this article at this web site. It’s perfect time to make some plans for
    the future and it’s time to be happy. I’ve read this post
    and if I could I wish to suggest you some interesting things or tips.
    Maybe you can write next articles referring to
    this article. I desire to read more things about it!

    It is appropriate time to make some plans for the future and it’s time
    to be happy. I have read this post and if I could I want to suggest you some interesting things or suggestions.
    Maybe you can write next articles referring to this article.

    I want to read even more things about it!

    http://foxnews.Co.uk/

    Suka

  5. The interest rates on these loans are very high, so prolonging your
    payments is not actually a good suggestion as you will find yourself paying again a lot
    more than you borrowed in the first occasion.

    Suka

  6. Hi there just wanted to give you a quick heads up and let you
    know a few of the images aren’t loading properly.
    I’m not sure why but I think its a linking issue.
    I’ve tried it in two different internet browsers and both show the same results.

    Suka

  7. What i don’t realize is in fact how you are not really a lot more neatly-preferred than you may be now.
    You’re so intelligent. You know therefore considerably on the subject of this topic,
    produced me in my view believe it from numerous various angles.

    Its like women and men don’t seem to be interested
    until it’s something to do with Girl gaga!
    Your personal stuffs outstanding. Always care for it up!

    Suka

  8. Howdy! I could have sworn I’ve been to this website before
    but after reading through some of the post I realized it’s new
    to me. Nonetheless, I’m definitely delighted I found it and I’ll be book-marking and
    checking back frequently!

    Suka

Tinggalkan Balasan

Isikan data di bawah atau klik salah satu ikon untuk log in:

Logo WordPress.com

You are commenting using your WordPress.com account. Logout /  Ubah )

Foto Google

You are commenting using your Google account. Logout /  Ubah )

Gambar Twitter

You are commenting using your Twitter account. Logout /  Ubah )

Foto Facebook

You are commenting using your Facebook account. Logout /  Ubah )

Connecting to %s