Verifying the Trigonometric Identity: ( cot^2x - cos^2x cos^2x cot^2x )

Trigonometric identities are a cornerstone of calculus and advanced mathematics. Understanding and verifying these identities is crucial for solving complex problems and simplifying expressions. In this guide, we will walk through the process of proving the identity ( cot^2x - cos^2x cos^2x cot^2x ).

Step-by-Step Solution

Let's carefully break down the given identity and demonstrate why it holds true.

Given Identity:

( cot^2x - cos^2x cos^2x cot^2x )

Left-Hand Side (LHS) Transformation:

Let's start by working on the left-hand side (LHS) of the equation and manipulate it step by step.

Step 1: Express ( cot^2x ) in terms of ( cos x ) and ( sin x ).

( cot^2x - cos^2x frac{cos^2x}{sin^2x} - cos^2x )

Step 2: Combine the terms over a common denominator.

( frac{cos^2x - cos^2x sin^2x}{sin^2x} )

Step 3: Simplify the numerator by factoring out ( cos^2x ).

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

Step 4: Use the Pythagorean identity ( 1 - sin^2x cos^2x ).

( frac{cos^2x cos^2x}{sin^2x} frac{cos^4x}{sin^2x} )

Step 5: Express ( frac{cos^4x}{sin^2x} ) in terms of ( cot x ).

( cos^2x cot^2x )

Note: ( cot x frac{cos x}{sin x} ) which means ( cot^2 x frac{cos^2 x}{sin^2 x} ).

Conclusion:

Since we have shown that the left-hand side simplifies to the same form as the right-hand side, the identity is indeed proven to be true.

Right-Hand Side (RHS) Verification:

The right-hand side (RHS) is already in the form ( cos^2x cot^2x ). Comparing it to the LHS, we observe that it matches our simplified form, confirming the identity.

Alternative Methods

There are multiple ways to simplify and prove the given identity. Here are a couple of additional approaches:

Method 1:

( cot^2x - cos^2x cos^2x cot^2x )

Using ( cot^2x frac{cos^2x}{sin^2x} ), we can simplify:

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

Method 2:

( cot^2x cos^2x - cos^2x cos^2x cot^2x )

Divide both sides by ( cos^2x cot^2x ):

( frac{cot^2x cos^2x - cos^2x}{cos^2x cot^2x} 1 )

Simplify the numerator:

( frac{cos^2x (1 - sin^2x)}{cos^2x cot^2x} 1 )

Using ( cot^2x frac{cos^2x}{sin^2x} ) and ( 1 - sin^2x cos^2x ):

( frac{cos^2x cos^2x}{cos^2x cos^2x / sin^2x} 1 )

Therefore, the identity is verified.

Conclusion

In this article, we have demonstrated multiple ways to verify the trigonometric identity ( cot^2x - cos^2x cos^2x cot^2x ). Understanding and mastering these methods are essential for tackling advanced mathematical problems involving trigonometric functions.