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Trigonometry Simplification | SMO, 2009 | Problem 26

Try this beautiful problem from Singapore Mathematics Olympiad, SMO, 2009 based on Trigonometry Simplification. You may use sequential hints.

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

Problem – Trigonometry Simplification (SMO Entrance)

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

Key Concepts

Trigonometry

Geometry

Challenges and Thrill – Pre College Mathematics

Try with Hints

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………………

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 ……………..

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