Exploring the Value of the Product of Squares of Cosines: cos^21° 3 cos^23° ... 179 cos^2179°

This article delves into the intricate process of calculating the value of the product of squares of cosines from 1 degree to 179 degrees. We will break down the solution step-by-step, applying mathematical identities and properties to find the final result. This exploration is particularly relevant to students and professionals in mathematics and related fields.

Introduction to the Problem

The problem at hand is to find the value of the expression:

S cos^21^circ 3 cos^23^circ 5 cos^25^circ ... 179 cos^2179^circ

Step-by-Step Solution

Let's begin by breaking down the problem into manageable steps.

Step 1: Recognizing Symmetry

A key observation is that the cosine function has a symmetry property:

cos^2x cos^2(180^circ - x)

This symmetry allows us to pair terms in the sum systematically. For each k from 1 to 89, we have:

cos^2(180^circ - 2k^circ) cos^2(2k^circ)

This means that:

cos^21^circ cos^2179^circ 2cos^21^circ
cos^23^circ cos^2177^circ 2cos^23^circ
...
cos^289^circ cos^22^circ 2cos^289^circ

Step 2: Rewriting the Sum

Since we can pair each term with its symmetric counterpart, the sum can be rewritten as:

S sum_{k1}^{89} 2k-1 cos^22k-1^circ

Step 3: Utilizing the Properties of Cosine

To simplify further, we can use the identity for cos^2:

cos^2x 1/2(1 cos2x)

Applying this identity, we rewrite the sum as:

S sum_{k1}^{89} 2k-1 1/2 (1 cos2(2k-1)^circ)/2

This can be split into two sums:

S 1/2 sum_{k1}^{89} 2k-1 1/4 sum_{k1}^{89} 2k-1 cos2(2k-1)^circ

Step 4: Calculate Each Part

First Sum:
The first sum can be calculated using a well-known formula for the sum of the first n odd numbers:

sum_{k1}^{n} 2k - 1 n^2

For n 89, we get:

sum_{k1}^{89} 2k - 1 89^2 7921

Therefore:

1/2 sum_{k1}^{89} 2k - 1 7921/2 3960.5

Second Sum:
The second sum involves the cosine function. Due to the oscillatory nature of cosine and symmetry, the contributions from positive and negative values will likely cancel out. Thus, this sum can be approximated to be 0.

Final Value

Combining the results from the two sums, we find:

S ≈ 3960.5 0 3960.5

Conclusion

The value of the product of squares of cosines from 1 degree to 179 degrees is approximately 3960.5. This exploration highlights the power of mathematical identities and symmetry in simplifying complex expressions.