What is the NO-SHORTCUT approach for learning great Mathematics?

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Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2009 based on Trigonometry Simplification.

If \(\frac {cos 100^\circ}{1-4 sin 25^\circ cos 25^\circ cos 50^\circ} = tan x^\circ \)

Find \( x^\circ \) ?

- 12
- 95
- 46
- 28

Trigonometry

Geometry

But try the problem first...

Answer: 95

Source

Suggested Reading

Singapore Mathematical Olympiad

Challenges and Thrill - Pre College Mathematics

First hint

If you really got stuck into this sum we can start from here

\(\frac {cos 100^\circ}{1-4 sin 25^\circ cos 25^\circ cos 50^\circ}\)

= \(\frac {cos 100^\circ}{1-2sin 50^\circ cos 50^\circ}\)

Now let's check with some basic values in trigonometry

\( Cos 2 A = cos^2 A - sin^2 A \) and

\(2 sin A cos A = sin 2 A\)

Now try the rest of the sum by using these two above mentioned values..................

Second Hint

Let's continue from the last hint :

\( cos 100^\circ = cos^2 50^\circ - sin^2 50^\circ \)

\( 2 sin 25^\circ cos 25^\circ = sin 50^\circ\)

\(\frac {cos^2 50^\circ - sin^2 50^\circ}{2sin 50^\circ cos 50^\circ}\)

\(\frac {cos^2 50^\circ - sin^2 50^\circ }{(cos 50^\circ - sin 50^\circ)^2}\)

Using \(a^2 - b^2 = (a+b) (a-b)\) formula

\(\frac {cos 50^\circ + sin 50^\circ}{cos 50^\circ - sin 50^\circ}\)

Do the rest of the steps .................

Final Step

Starting from right after the last hint:

\(\frac {cos 50^\circ + sin 50^\circ}{cos 50^\circ - sin 50^\circ}\)

= \(\frac {1+ tan 50^\circ}{1-tan 50^\circ}\)

= \(\frac {tan 45^\circ + tan 50^\circ}{1-tan 45^\circ tan 50 ^\circ}\)

= \( tan 95^\circ\) - Answer

What to do to shape your Career in Mathematics after 12th?

From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:

- What are some of the best colleges for Mathematics that you can aim to apply for after high school?
- How can you strategically opt for less known colleges and prepare yourself for the best universities in India or Abroad for your Masters or Ph.D. Programs?
- What are the best universities for MS, MMath, and Ph.D. Programs in India?
- What topics in Mathematics are really needed to crack some great Masters or Ph.D. level entrances?
- How can you pursue a Ph.D. in Mathematics outside India?
- What are the 5 ways Cheenta can help you to pursue Higher Mathematics in India and abroad?

Cheenta has taken an initiative of helping College and High School Passout Students with its "Open Seminars" and "Open for all Math Camps". These events are extremely useful for students who are really passionate for Mathematic and want to pursue their career in it.

Cheenta is a knowledge partner of Aditya Birla Education Academy

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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