Mathematics is a field rich in identities and equations. One interesting identity revolves around the product of cosines. Let us delve into proving the following identity:

Proving the Cosine Product Identity

Let us begin with the left-hand side (LHS) of the equation:

LHS cosAcos2Acos4Acos8A

Step 1: Transforming the Left-Hand Side

Multiplying and dividing the LHS by 2sinA:

LHS cosAcos2Acos4Acos8A * (frac{2sinA}{2sinA})

LHS (frac{1}{2}sinA[2sinAcosAcos2Acos4Acos8A])

LHS (frac{1}{2}sinA[2sin2Acos2Acos4Acos8A])

LHS (frac{1}{4}sinA[2sin2Acos2Acos4Acos8A])

LHS (frac{1}{4}sinA[2sin4Acos4Acos8A])

LHS (frac{1}{4}sinA[2sin4Acos4Acos8A])

LHS (frac{1}{8}sinA[2sin8Acos8A])

LHS (frac{1}{16}sinA[2sin16A])

LHS (frac{sin16A}{16sinA})

Step 2: Simplifying the Expression

Recall the identity sin2x (frac{1-cos2x}{2}) and use it to simplify the expression further:

16 24

sin16A sin(2*8A) 2sin8Acos8A

sin28A (sin8Acos8A)2 sin28A

sin28A * sin28A sin48A sin16A

Cross-Verification

Let's cross-verify the identity with a specific value. For example, let A 60°.

LHS cos60°cos120°cos240°cos480° cos60° (frac{1}{2}) cos120° (-frac{1}{2}) cos240° (-frac{1}{2}) cos480° (frac{1}{2}) LHS (frac{1}{2} * -frac{1}{2} * -frac{1}{2} * frac{1}{2} -frac{1}{16})

Conclusion

The left-hand side simplifies to -(frac{1}{16}).

RHS (frac{sin^4(2A)}{2^4sinA} frac{1}{16sinA})

For A 60°, RHS (frac{9}{128sqrt{3}})

Therefore, the two sides do not match for A 60°, indicating that the identity is not correct for all values of A.

**Note:** The provided expression seems to be an identity rather than an equation to be solved. The verification process shows that the given identity is not universally true.

Happy Maths!