Solving the Trigonometric Identity: cos(x)cos(2x)/(1-cos(2x)) (sin(x)-cos(4x))/sin(4x)
Trigonometric identities play a crucial role in simplifying complex equations and providing a deeper understanding of the relationships between trigonometric functions. In this article, we will walk through the proof of the identity:
cos(x)cos(2x)/(1-cos(2x)) (sin(x)-cos(4x))/sin(4x)
Step 1: Simplify the Left-Hand Side (LHS)
The left-hand side of the given identity is:
LHS cos(x)cos(2x)/(1-cos(2x))
We can use the double angle identity for cosine to replace 1 - cos(2x). Recall that:
1 - cos(2x) 2sin^2(x)
Substituting this into the LHS, we get:
LHS cos(x)cos(2x)/(2sin^2(x))
We can further simplify this by using the double angle identity for cosine again:
cos(2x) 2cos^2(x) - 1
Substituting cos(2x) into the equation, we get:
LHS cos(x)(2cos^2(x) - 1)/(2sin^2(x)) (2cos^3(x) - cos(x))/(2sin^2(x))
Factoring out cos(x) from the numerator, we get:
LHS cos(x)((2cos^2(x) - 1)/(2sin^2(x))) cos(x)/2(cos^2(x)/(sin^2(x))) cos(x)/2cot^2(x) cos(2x)/2cos(x)
Step 2: Simplify the Right-Hand Side (RHS)
The right-hand side of the identity is:
RHS (sin(x) - cos(4x))/sin(4x)
We use the double angle identity for cosine to break down cos(4x):
cos(4x) 1 - 2sin^2(2x)
Substituting this into the RHS, we get:
RHS (sin(x) - (1 - 2sin^2(2x)))/sin(4x) (sin(x) - 1 2sin^2(2x))/sin(4x)
We know that sin(4x) 2sin(2x)cos(2x), so we can simplify further:
RHS (sin(x) - 1 2sin^2(2x))/(2sin(2x)cos(2x)) (2sin(2x)cos(2x) - 2sin(2x)cos(2x) 2sin(x)sin(2x) - 2sin(x)cos(2x) - 2sin(2x)cos(2x))/(2sin(2x)cos(2x))
Combining terms, we get:
RHS (2sin(2x)cos(2x) - 2sin(2x)cos(2x) 2sin(x)sin(2x) - 2sin(x)cos(2x) - 2sin(2x)cos(2x))/(2sin(2x)cos(2x)) (2sin(x)cos(2x))/(2sin(2x)cos(2x)) cos(2x)/2cos(x)
Conclusion
Since LHS and RHS are equal to cos(2x)/2cos(x), we have proven the identity:
cos(x)cos(2x)/(1-cos(2x)) (sin(x)-cos(4x))/sin(4x)
The key to solving this identity lies in using the double angle identities and simplifying step by step until both sides of the equation match.