Simplifying Trigonometric Expressions: A Step-by-Step Guide

In this guide, we will explore how to simplify a complex trigonometric expression using basic identities and algebraic techniques. This process is not only crucial for solving advanced mathematical problems but also a foundational concept for understanding calculus and higher mathematics.

Understanding the Problem

The given expression is:

frac;{cos x sin x}{sin x cos^2 x sin^2 x} frac{cos x}{cos^2 x sin^2 x}

Our goal is to simplify this expression to its simplest form, which is cos x.

Step-by-Step Solution

Step 1: Simplify the Denominator

First, we factor out sinx from the denominator:

When we simplify the denominator by taking out sinx, we are left with:

frac{cos x sin x}{sin x cos^2 x sin^2 x} frac{cos x}{cos^2 x sin x}

Step 2: Use Trigonometric Identities

Next, we use the trigonometric identity:

sin^2x cos^2x 1

Substituting this identity into our expression, we get:

frac{cos x}{cos^2 x sin^2 x} frac{cos x}{1}

This simplifies to:

cos x

Verification

To verify our solution, we can set up the problem using a function notation:

Let f(x) frac{cos x sin x}{sin x cos^2 x sin^3 x}

Factoring out sinx in the denominator:

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

This can be further simplified to:

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

Using the identity sin^2 x cos^2 x 1 again:

f(x) frac{cos x}{1} cos x

Conclusion

We have successfully simplified the given trigonometric expression to its simplest form using fundamental algebraic and trigonometric principles. This process not only demonstrates the power of trigonometric identities but also serves as a valuable exercise in mathematical proof.

Key Takeaways:

Factor out common terms in the numerator and denominator to simplify the expression. Use trigonometric identities to substitute and simplify further.

By following these steps, you can solve similar problems and develop a stronger foundation in trigonometry and algebra.