Proving Trigonometric Identity: tan^2 x - 1 ÷ (1 / tan^2 x) 1 - 2 cos^2 x

Understanding and proving trigonometric identities is a fundamental skill in mathematics, essential for solving complex problems in calculus, physics, and engineering. One common identity is to prove that tan^2 x - 1 ÷ (1 / tan^2 x) 1 - 2 cos^2 x. This article will guide you through the proof with detailed steps and explanations.

Introduction to the Trigonometric Expression

To prove this identity, we start by manipulating the left-hand side (LHS) of the equation. The expression is:

(dfrac{tan^2 x - 1}{dfrac{1}{tan^2 x}})

This expression is initially challenging to simplify directly. However, by breaking down the components, we can make the simplification clearer.

Step-by-Step Proof

Step 1: Rewrite the Expression Let's first rewrite the original expression using the definition of tangent and sine:

(dfrac{tan^2 x - 1}{dfrac{1}{tan^2 x}} dfrac{dfrac{sin^2 x}{cos^2 x} - 1}{dfrac{1}{tan^2 x}} dfrac{dfrac{sin^2 x}{cos^2 x} - 1}{dfrac{cos^2 x}{sin^2 x}})

Step 2: Simplify the Denominator and Numerator To simplify this, we need to manipulate both the numerator and the denominator. Multiply the numerator and denominator by (sin^2 x), which gives us:

(dfrac{sin^2 x left(dfrac{sin^2 x}{cos^2 x} - 1right)}{cos^2 x} dfrac{sin^2 x - cos^2 x}{cos^2 x})

Step 3: Use the Pythagorean Identity We know that (sin^2 x cos^2 x 1). Applying this identity to the numerator, we get:

(dfrac{1 - cos^2 x - cos^2 x}{cos^2 x} dfrac{1 - 2cos^2 x}{cos^2 x})

Step 4: Simplify Further This simplifies to:

(1 - 2cos^2 x)

Thus, the left-hand side (LHS) of the original equation simplifies to:

(1 - 2cos^2 x)

This is identical to the right-hand side (RHS) of the given identity. Hence, we have proven the identity:

(tan^2 x - 1 div left(1 / tan^2 xright) 1 - 2cos^2 x)

Conclusion

Understanding how to prove trigonometric identities is crucial for advanced mathematical studies. The proof provided here demonstrates the manipulation of trigonometric functions to show their equivalence, a skill that will be invaluable for more complex problem-solving.