Problem 81 - Algebra and Trigonometry

Question

\(73-90\) Prove the identity. $$ \tan 3 x=\frac{3 \tan x-\tan ^{3} x}{1-3 \tan ^{2} x} $$

Step-by-Step Solution

Answer

The identity is proven by using the tangent triple angle formula.

Step 1: Understand the Identity

The problem asks us to prove the trigonometric identity \(\tan 3x = \frac{3 \tan x - \tan^3 x}{1 - 3 \tan^2 x}\). This means we need to show that the left-hand side (LHS) and the right-hand side (RHS) are equivalent.

Step 2: Use the Tangent Triple Angle Formula

Recall the tangent triple angle identity: \(\tan(3x) = \frac{3\tan(x) - \tan^3(x)}{1 - 3\tan^2(x)}\). This is exactly the RHS of our identity to be proven, so we need to verify that both sides follow from established trigonometric identities.

Step 3: Express Tangent in Terms of Sine and Cosine

First, use the definition of tangent: \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). Then explore how to express \(\tan(3x)\) in terms of \(\sin(3x)\) and \(\cos(3x)\).

Step 4: Use Trigonometric Angle Formulas

Use the established identities:\[\sin(3x) = 3\sin(x) - 4\sin^3(x)\]\[\cos(3x) = 4\cos^3(x) - 3\cos(x)\]These are derived from angle duplication and reduction formulas.

Step 5: Substitute and Simplify

Substitute the expressions for \(\sin(3x)\) and \(\cos(3x)\) into the expression for \(\tan(3x)\):\[\tan(3x) = \frac{3\sin(x) - 4\sin^3(x)}{4\cos^3(x) - 3\cos(x)}\]Simplify this using the identity \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), leading to:\[\tan(3x) = \frac{3\frac{\sin(x)}{\cos(x)} - \left(\frac{\sin(x)}{\cos(x)}\right)^3}{1 - 3\left(\frac{\sin(x)}{\cos(x)}\right)^2}\]

Step 6: Simplify and Equivalence Check

Notice that simplification leads directly to:\[\tan(3x) = \frac{3\tan(x) - \tan^3(x)}{1 - 3\tan^2(x)}\]This matches our initially given RHS, proving that each side of the identity is equivalent, thereby verifying the identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable where both sides are defined. These identities can simplify complex trigonometric expressions. For example, one of the core identities is the Pythagorean identity:

• \( an^2(x) + 1 = rac{1}{ ext{cos}^2(x)}\)

These identities allow you to express one trigonometric function in terms of another. This is crucial when proving other identities or simplifying expressions. Having a good grasp of these identities can significantly reduce the complexity of problems, particularly when dealing with compound angles or multiple angle scenarios. It's important to familiarize yourself with identities like the angle sum and difference identities, double angle formulas, and more to see how they interlock and assist in proving more intricate relationships like the tangent triple angle formula.

Trigonometric Proofs

Trigonometric proofs involve demonstrating that two expressions are equivalent using known trigonometric identities. In a proof, it's essential to clearly show each transformation or substitution step-by-step to avoid any logical jumps.
One effective method in proofs is to express tangent or any trigonometric functions using sine and cosine, as these are the more basic trigonometric functions. This strategy allows you to leverage well-known identities in a straightforward way.

• Start by transforming the expression on one side using known identities.
• Next, manipulate the expression to match the other side.
• If necessary, simplify complex fractions or factor expressions.

This step-by-step logical breakdown leads us to align the left-hand side of the expression with the right-hand side, thus verifying the identity. Patience and practice in laying out each piece explicitly are key to mastering these proofs.

Angle Formulas

Angle formulas in trigonometry help calculate the trigonometric functions for more complex angles, like when a simple angle is multiplied by a factor (as in the tangent triple angle formula). They also encompass calculations for angles that are sums or differences of other angles.
For instance, the formulas for \(\sin(3x)\) and \(\cos(3x)\) are derived from the angle addition formulas and are fundamental when dealing with expressions like the tangent triple angle formula:

• \(\sin(3x) = 3\sin(x) - 4\sin^3(x)\)
• \(\cos(3x) = 4\cos^3(x) - 3\cos(x)\)

These come from iteratively applying the double angle formulas. Understanding how to use these to break down and rebuild angles not only aids in proving identities but also helps in solving equations involving trigonometric functions. It's through these fundamental relations and repeated proofs that you can deepen your understanding of trigonometry and extend your problem-solving toolbox.