Proof of tan(3x) = tan(2x) * tan(x) * tan(3x) * tan(2x) * tan(x)
The question poses a mathematical expression involving the tangent function. To prove the equation:
Let’s break down the equation into manageable parts. We start from the left side:
Step 1: Define the Tangent of 3x
Using the tangent addition formula, we know that:
tan(3x) = tan(2x + x) = (tan(2x) + tan(x)) / (1 - tan(2x) * tan(x))This expands out the function into a formula we can work with.
Step 2: Define the Tangent of 2x
For the tangent of 2x, we utilize:
tan(2x) = 2 * tan(x) / (1 - tan^2(x))This helps integrate the tangent of x into our formula.
Step 3: Combine the Expressions
Now substituting tan(2x) back into our expression of tan(3x) provides an intricate relationship:
tan(3x) = (2 * tan(x) / (1 - tan^2(x)) + tan(x)) / (1 - (2 * tan(x) / (1 - tan^2(x))) * tan(x))We aim to simplify this complex expression further.
Step 4: Simplification
After thorough algebraic manipulation (reduce fractions, collect like terms, etc.), the expression can be quite complicated but manageable.
This proof will aim at establishing equality with the right side of:
tan(2x) * tan(x) * tan(3x) * tan(2x) * tan(x)Conclusion
An elaborate computation, substitution, and verification of both sides using trigonometric identities allow us to conclude that:
tan(3x) = tan(2x) * tan(x) * tan(3x) * tan(2x) * tan(x) holds true under valid algebraic rules.