Proving Trigonometric Identities: tan^2 x sin^2 x tan^2 x - sin^2 x

This article explains how to prove the trigonometric identity tan^2 x sin^2 x tan^2 x - sin^2 x. We will use the definitions and properties of trigonometric functions such as sine and tangent, along with some fundamental identities. Let's break it down step-by-step.

Proof of the Identity

To prove the identity tan^2 x sin^2 x tan^2 x - sin^2 x, we start by using the definitions of the trigonometric functions involved. Recall that the tangent function is defined as:

[tan x frac{sin x}{cos x}].

This implies that:

[tan^2 x frac{sin^2 x}{cos^2 x}].

Now substitute this into the original equation:

Left-Hand Side (LHS)

Let's start with the left-hand side (LHS) of the equation:

[text{LHS} tan^2 x sin^2 x left(frac{sin^2 x}{cos^2 x}right) sin^2 x frac{sin^4 x}{cos^2 x}].

Right-Hand Side (RHS)

Now consider the right-hand side (RHS) of the equation:

[text{RHS} tan^2 x - sin^2 x frac{sin^2 x}{cos^2 x} - sin^2 x].

To combine these terms, we need a common denominator:

[text{RHS} frac{sin^2 x}{cos^2 x} - frac{sin^2 x cos^2 x}{cos^2 x} frac{sin^2 x - sin^2 x cos^2 x}{cos^2 x}].

Factor out [sin^2 x] in the numerator:

[text{RHS} frac{sin^2 x (1 - cos^2 x)}{cos^2 x}].

Using the Pythagorean identity [1 - cos^2 x sin^2 x]:

[text{RHS} frac{sin^2 x cdot sin^2 x}{cos^2 x} frac{sin^4 x}{cos^2 x}].

Now we can compare both sides:

[text{LHS} frac{sin^4 x}{cos^2 x}]

[text{RHS} frac{sin^4 x}{cos^2 x}].

Since both sides are equal, we have proven the original identity:

[tan^2 x sin^2 x tan^2 x - sin^2 x].

Alternative Approach

Let's begin the proof with the right-hand side (RHS) of the identity:

[text{RHS} frac{sin^2 x}{cos^2 x} - sin^2 x].

Recall:

[tan x frac{sin x}{cos x}], so

[tan^2 x frac{sin^2 x}{cos^2 x}].

Thus:

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

Combine the terms with a common denominator:

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

Factor out [sin^2 x] in the numerator:

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

Using the Pythagorean identity [sin^2 x 1 - cos^2 x]:

[ frac{sin^2 x cdot sin^2 x}{cos^2 x} frac{sin^4 x}{cos^2 x}].

Hence:

[text{RHS} frac{sin^4 x}{cos^2 x}].

Similarly, we have:

[text{LHS} left(frac{sin^2 x}{cos^2 x}right) sin^2 x frac{sin^4 x}{cos^2 x}].

Thus, we have proved the identity:

[tan^2 x sin^2 x tan^2 x - sin^2 x].

Conclusion

In this article, we have demonstrated the proof of the trigonometric identity tan^2 x sin^2 x tan^2 x - sin^2 x. By utilizing the basic definitions and identities of sine, cosine, and tangent, we were able to show that the left-hand side and the right-hand side of the equation are equal.

Understanding and mastering these trigonometric identities is crucial in various mathematical and engineering applications. If you have any more questions or need further clarification on this topic, feel free to reach out.