Proving Trigonometric Identities: A Comprehensive Guide

Trigonometric identities are a fascinating area in mathematics, allowing us to simplify complex expressions and prove various relationships between trigonometric functions. In this article, we will delve into the detailed proof of the identity: (frac{1 - sin^4 x}{cos^4 x} 1 2 tan^2 x).

Proving the Identity

Let's start by breaking down the left-hand side of the given identity:

Step 1: Simplify the Left-Hand Side

We can rewrite the expression (1 - sin^4 x) using the difference of squares:

[1 - sin^4 x (1 - sin^2 x)(1 sin^2 x)]

We know that (1 - sin^2 x cos^2 x), so we can substitute this into the expression:

[(1 - sin^2 x)(1 sin^2 x) cos^2 x(1 sin^2 x)]

Step 2: Substitute Back into the Equation

Substituting this result back into the original expression, we get:

[frac{1 - sin^4 x}{cos^4 x} frac{cos^2 x(1 sin^2 x)}{cos^4 x} frac{1 sin^2 x}{cos^2 x}]

Step 3: Rewrite in Terms of (tan^2 x)

Now, let's rewrite the expression (frac{1 sin^2 x}{cos^2 x}) in terms of (tan^2 x):

[frac{1 sin^2 x}{cos^2 x} frac{1}{cos^2 x} frac{sin^2 x}{cos^2 x}] [frac{1}{cos^2 x} sec^2 x] [frac{sin^2 x}{cos^2 x} tan^2 x]

So, the expression becomes:

[sec^2 x tan^2 x]

Using Pythagorean Identities

Recall the Pythagorean identity: (sec^2 x 1 tan^2 x). Substituting this into our expression, we get:

[1 tan^2 x tan^2 x 1 2tan^2 x]

This completes the proof:

[frac{1 - sin^4 x}{cos^4 x} 1 2 tan^2 x]

Thus, the identity is proven. Let's confirm this step-by-step through another method:

Another Method of Proving the Identity

Starting with the left-hand side:

[frac{1 - sin^4 x}{cos^4 x}]

We can express the numerator as a difference of two squares:

[frac{1 - (sin^2 x)^2}{cos^4 x} frac{(1 - sin^2 x)(1 sin^2 x)}{cos^4 x}]

Using the Pythagorean identity (cos^2 x 1 - sin^2 x), we get:

[frac{cos^2 x(1 sin^2 x)}{cos^4 x} frac{1 sin^2 x}{cos^2 x}]

Next, we rewrite this expression in terms of (tan^2 x):

[frac{1 sin^2 x}{cos^2 x} frac{1}{cos^2 x} frac{sin^2 x}{cos^2 x} sec^2 x tan^2 x]

Finally, using (sec^2 x 1 tan^2 x), we have:

[1 tan^2 x tan^2 x 1 2tan^2 x]

Therefore, the identity is proven. Q.E.D.

Through these detailed steps, we have shown how to prove the given trigonometric identity using both methods. This exercise not only solidifies the understanding of trigonometric identities but also strengthens the skills in handling algebraic manipulations and applying known identities.