Proving the Trigonometric Identity: TanX / (SecX - 1) (SecX 1) / TanX

This guide aims to help you verify a given trigonometric identity, namely that TanX / (SecX - 1) equals (SecX 1) / TanX. We will go through a step-by-step process to show this equality using different methods.

Method 1: Step-by-Step Simplification

Let's start by recalling the definitions of tanX and secX.

Step 1: Rewrite tanX and secX

[tanX frac{sinX}{cosX}] [secX frac{1}{cosX}]

Step 2: Substitute the Definitions

Substitute these definitions into the left side:

[frac{tanX}{secX - 1} frac{frac{sinX}{cosX}}{frac{1}{cosX} - 1}]

Step 3: Simplify the Denominator

The denominator becomes:

[frac{1}{cosX} - 1 frac{1 - cosX}{cosX}]

So, we have:

[frac{tanX}{secX - 1} frac{frac{sinX}{cosX}}{frac{1 - cosX}{cosX}} frac{sinX}{1 - cosX}]

Step 4: Simplify the Right Side

Now, simplify the right side:

[frac{secX 1}{tanX} frac{frac{1}{cosX} 1}{frac{sinX}{cosX}}]

The numerator becomes:

[frac{1 cosX}{cosX}]

Thus, the right side simplifies to:

[frac{frac{1 cosX}{cosX}}{frac{sinX}{cosX}} frac{1 cosX}{sinX}]

Step 5: Simplify Using Trigonometric Identities

Now let's simplify both sides to show they are equal:

[frac{sinX}{1 - cosX}]

[frac{1 cosX}{sinX}] Now, multiply the left side by (frac{1 cosX}{1 cosX}):

[frac{sinX}{1 - cosX} times frac{1 cosX}{1 cosX} frac{sinX(1 cosX)}{(1 - cosX)(1 cosX)}] Knowing that:

[(1 - cosX)(1 cosX) 1 - cos^2X sin^2X] So, we have:

[frac{sinX(1 cosX)}{sin^2X} frac{1 cosX}{sinX}]

This shows that both expressions are indeed equal.

Method 2: Another Approach Using Trigonometric Identities

We can also verify the identity by using the Pythagorean identity (sin^2X cos^2X 1).

Step 6: Simplify the Right Side Using Identities

[tan^2x sec^2x - 1 secx 1secx - 1] Thus, we have:

[frac{tanx}{secx - 1} frac{secx 1}{tanx}]

Verification:

Another way to verify the identity is to use the identity (sin^2 x - cos^2 x 1) and rewrite it as:

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

[tan^2 x - 1 sec^2 x]

[tan^2 x sec^2 x - 1]

[tan^2x sec x 1 sec x - 1]

[frac{tan x}{sec x - 1} frac{sec x 1}{tan x}quadcheckmark]

Conclusion

We have shown that through different methods, the trigonometric identity (frac{tanX}{secX - 1} frac{secX 1}{tanX}) holds true. This verifies that both expressions are indeed equal.

Key Takeaways: Understanding the definitions and properties of trigonometric functions is crucial. Using trigonometric identities effectively can help in simplifying complex expressions. Careful algebraic manipulation and cross-multiplication will verify the identity.