Proving the Trigonometric Identity: 2 cos^2 x / (1 - cos^2 x) 2 / tan^2 x

Introduction:

Trigonometric identities are fundamental in mathematics, providing powerful tools to simplify expressions and solve equations. One such identity is proving the following expression:

2 cos^2 x / (1 - cos^2 x) 2 / tan^2 x

Step-by-Step Proof

Let's start by breaking down the left-hand side (LHS) of the equation and transforming it into the form of the right-hand side (RHS).

LHS

The left-hand side of the equation is 2 cos^2 x / (1 - cos^2 x).

Step 1: Substitute the Value of 1 - cos^2 x

First, we will use the Pythagorean identity, which states that sin^2 x cos^2 x 1. By rearranging this identity, we can understand that 1 - cos^2 x sin^2 x.

This allows us to rewrite the LHS:

2 cos^2 x / sin^2 x

Step 2: Express in Terms of Tangent

Next, we use the definition of the tangent function: tan x sin x / cos x. Taking the reciprocal, we get 1/tan x cos x / sin x.

Substitute this into our expression:

2 * (1/tan x) ^ 2

Thus, the left-hand side simplifies to:

2 / tan^2 x

This is exactly the same as the right-hand side (RHS), so we have successfully proved the identity.

Verification

To further ensure the accuracy of our proof, let's verify the identity step-by-step:

LHS

Starting with the initial expression:

2 cos^2 x / (1 - cos^2 x)

Substitute the value of 1 - cos^2 x:

2 cos^2 x / sin^2 x

Divide both the numerator and denominator by cos^2 x:

2 / (tan^2 x) 2 / tan^2 x

This matches the RHS, confirming the identity.

Conclusion

In this article, we have demonstrated a detailed proof of the trigonometric identity: 2 cos^2 x / (1 - cos^2 x) 2 / tan^2 x. This proof utilizes fundamental trigonometric identities and algebraic manipulation to simplify the expression and verify its equivalence to the target form.

Understanding and mastering such proofs is essential for students and professionals in mathematics and related fields. By practicing similar problems, one can enhance their skills in trigonometry and improve their ability to solve complex problems.

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