Proving the Identity: $frac{cos x cot x}{cos x - cot x} sec x - tan x$

Understanding and proving trigonometric identities is a fundamental aspect of mathematics, particularly in calculus and advanced algebra. This article provides a detailed step-by-step guide on how to prove the identity $frac{cos x cot x}{cos x - cot x} sec x - tan x$. This process will not only reinforce your understanding of trigonometric functions but also enhance your problem-solving skills.

Step-by-Step Guide to Proving Trigonometric Identities

To prove that $frac{cos x cot x}{cos x - cot x} sec x - tan x$, we will express all trigonometric functions in terms of sine and cosine and follow a systematic approach.

Step 1: Expressing Trigonometric Functions in Terms of Sine and Cosine

Recall the definitions of the trigonometric functions in terms of sine and cosine:

$cot x frac{cos x}{sin x}$ $sec x frac{1}{cos x}$ $tan x frac{sin x}{cos x}$

Step 2: Simplify the Left-Hand Side

Let's rewrite the left-hand side (LHS) of the identity:

[LHS frac{cos x cot x}{cos x - cot x}]

Substitute the definition of $cot x$:

[LHS frac{cos x left( frac{cos x}{sin x} right)}{cos x - left( frac{cos x}{sin x} right)}]

Simplify the numerator and the denominator:

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

The $sin x$ in the denominator and numerator of the fraction simplify:

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

Step 3: Simplify the Right-Hand Side

Now, let's rewrite the right-hand side (RHS) of the identity:

[RHS sec x - tan x]

Substitute the definitions of $sec x$ and $tan x$:

[RHS frac{1}{cos x} - frac{sin x}{cos x} frac{1 - sin x}{cos x}]

Find a common denominator and simplify:

[RHS frac{1 - sin x}{cos x}]

Step 4: Show That LHS RHS

From the previous steps, we have:

[LHS frac{cos x}{sin x - 1}]

And:

[RHS frac{1 - sin x}{cos x}]

To show that LHS RHS, we can manipulate LHS to match RHS:

[LHS frac{cos x}{sin x - 1} frac{cos x (1 - sin x)}{cos x (sin x - 1)} frac{cos x (1 - sin x)}{cos x (1 - sin x)}] [LHS frac{1 - sin x}{cos x} RHS]

Since both sides are equal, we conclude:

[frac{cos x cot x}{cos x - cot x} sec x - tan x]

Conclusion

The detailed steps provided above prove the given trigonometric identity. This process reinforces the importance of expressing trigonometric functions in terms of sine and cosine, and using algebraic manipulation to simplify and equate sides of an equation. Understanding and practicing these steps will enhance your trigonometric problem-solving skills.