Proving the Trigonometric Identity: tan^2 x - sin^2 x tan^2 x sin^2 x
Introduction
Understanding and proving trigonometric identities is a fundamental aspect of trigonometry. One such identity is (tan^2 x - sin^2 x tan^2 x sin^2 x). This article will provide multiple methods to prove this identity, making use of basic trigonometric transformations.
Method 1: Initial Substitution and Simplification
Consider the given identity: (tan^2 x - sin^2 x tan^2 x sin^2 x).
The left-hand side (LHS) can be rewritten as:
(begin{aligned} tan^2 x - sin^2 x frac{sin^2 x}{cos^2 x} - sin^2 x sin^2 x left(frac{1}{cos^2 x} - 1right) sin^2 x left(frac{1 - cos^2 x}{cos^2 x}right) sin^2 x left(frac{sin^2 x}{cos^2 x}right) sin^2 x tan^2 x end{aligned})
This shows that the LHS equals the right-hand side (RHS), (tan^2 x sin^2 x). Thus, the identity is proven.
Method 2: Detailed Step-by-Step Proof
Let's go through the proof step-by-step:
Start with the left-hand side (LHS): (tan^2 x - sin^2 x). Substitute the trigonometric identity for (tan^2 x) as (frac{sin^2 x}{cos^2 x}):(tan^2 x - sin^2 x frac{sin^2 x}{cos^2 x} - sin^2 x)
Factor out (sin^2 x):(sin^2 x left(frac{1}{cos^2 x} - 1right))
Simplify the expression inside the parentheses:(sin^2 x left(frac{1 - cos^2 x}{cos^2 x}right))
Recall the Pythagorean identity: (1 - cos^2 x sin^2 x). Substituting this in:(sin^2 x left(frac{sin^2 x}{cos^2 x}right))
Recognize that (tan^2 x frac{sin^2 x}{cos^2 x}). Therefore:(sin^2 x tan^2 x)
Thus, the LHS equals the RHS: (tan^2 x sin^2 x).The identity is proven.
Another Proof via Substitution
Consider the identity in another form: (tan^2 x - sin^2 x).
Let (a tan^2 x) and (b sin^2 x). Then, the left-hand side (LHS) can be written as:
(a - b ab)
With (a tan^2 x) and (b sin^2 x), this becomes:
(tan^2 x - sin^2 x tan^2 x sin^2 x)
This confirms that the identity holds.
Conclusion
The provided identity, (tan^2 x - sin^2 x tan^2 x sin^2 x), can be proven in multiple ways. Each method involves the use of fundamental trigonometric identities and algebraic manipulation. Understanding these methods helps in manipulating and proving other trigonometric identities as well.