The value of cos(π/4) ⋅ cos(π/8) ⋅ cos(π/16) ... cos(π/2ⁿ) equals

• 1/2ⁿ cosec(π/2_n)
• 1/2ⁿ cosec(π/2_n - 1)
• 1/2ⁿ cosec(π/2_n)
• 1/2ⁿ cosec(π/2_n)

The expression you provided involves a product of cosine functions at specific angles. To understand the value of this product, we can break it down step by step.

Understanding the Product of Cosines

The product is given by:

• cos(π/4)
• cos(π/8)
• cos(π/16)
• ... up to cos(π/2ⁿ)

Formula for the Product

There is a known result for the product of cosines:

cos(π/4) ⋅ cos(π/8) ⋅ cos(π/16) ⋅ ... ⋅ cos(π/2ⁿ) = 1/2ⁿ cosec(π/2ⁿ)

Breaking Down the Result

This means that as you multiply these cosine values, the result simplifies to:

• 1/2ⁿ: This indicates how the product decreases as n increases.
• cosec(π/2ⁿ): This function is the reciprocal of sine, which plays a crucial role in trigonometric identities.

Final Expression

Thus, the complete expression can be summarized as:

Value = 1/2ⁿ cosec(π/2ⁿ)

This relationship highlights the connection between the product of cosines and the cosecant function, providing a neat and elegant result in trigonometry.