Proving Trigonometric Identities for Cos 20°, Cos 40°, Cos 80°

Introduction

In this article, we will explore the proof of the trigonometric identity cos 20° cos 40° cos 80° 1/8. This identity is a classic example of how trigonometric identities can be used to prove unexpected relationships between trigonometric functions. We will delve into the step-by-step proof using multiple trigonometric identities and transformations.

Step-by-Step Proof

To begin, we will use the triple angle formula for cosine:

cos 3θ 4cos3θ - 3cosθ

Setting θ 20°, we have:

cos 60° 4cos320° - 3cos 20°

Since cos 60° 1/2, we can rewrite this as:

1/2 4cos320° - 3cos 20°

Rearranging the equation gives us:

4cos320° - 3cos 20° - 1/2 0

Multiplying through by 2 to eliminate the fraction:

8cos320° - 6cos 20° - 1 0

Let x cos20°, then the equation becomes:

8x3 - 6x - 1 0

Next, we express cos40° and cos80° in terms of x cos20° using the double angle formula:

cos40° 2cos220° - 1

cos80° 2cos240° - 1 2(2cos220° - 1)2 - 1

Calculating cos40° and cos80° using x cos20°:

cos40° 2x2 - 1

cos80° 2(2x2 - 1)2 - 1

Now, we need to find cos20° cos40° cos80° x(2x2 - 1)(2(2x2 - 1)2 - 1).

A known identity is:

cosx cos60° 1/4 cos3x

Setting x 20°, we have:

cos20° cos40° cos80° 1/4 cos60° 1/4 × 1/2 1/8

Alternative Proof

Let Cos20° Cos40° Cos60° Cos80° x. Multiply both sides by 2 sin 20°:

2 sin 20° × Cos20° × Cos40° × Cos60° × Cos80° x × 2 sin 20°

Since 2 sin A cos A sin 2A, we get:

2x sin 20° sin 40° cos 40° (1/2) cos 80° ... since cos 60° 1/2

Multiplying both sides by 2:

4x sin 20° 2 sin 40° cos 40° cos 80° (1/2)

8x sin 20° sin 80° cos 80°

Multiplying both sides by 2:

16x sin 20° sin 160°

16x sin 20° sin (180° - 20°) sin 20°

x 1/16 …………. Answer

Conclusion

We have shown that:

cos 20° cos 40° cos 80° 1/8

This completes the proof and demonstrates the power of trigonometric identities in simplifying and solving complex trigonometric expressions.