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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

But try the problem first…

Source

Challenges and Thrill – Pre College Mathematics

## Try with Hints

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