Verify the identity: tan 2x + sec^2x = 1 + 2tan^2x

Verify the identity: tan 2x + sec^2x = 1 + 2tan^2x Which of the following four statements establishes the identity? A. tan^2x + sec^2x = tan^2x + (1 – tan^2x) = 1 + 2tan^2x B. tan^2x + sec^2x = tan^2x + tan^2x = 1 + 2tan^2x C. tan^2x + sec^2x = tan^2x + (tan^2x – 1) = 1 + 2tan^2x D. tan^2x + sec^2x = 1 = 1 + 2tan^2x

The Correct Answer and Explanation is:

The correct answer is Statement B.

We are asked to verify the identity:

tan⁡2x+sec⁡2x=1+2tan⁡2x\tan^2 x + \sec^2 x = 1 + 2 \tan^2 x

To do this, we recall the fundamental Pythagorean identity involving tangent and secant:

sec⁡2x=1+tan⁡2x\sec^2 x = 1 + \tan^2 x

Substituting this identity into the left-hand side of the equation:

tan⁡2x+sec⁡2x=tan⁡2x+(1+tan⁡2x)\tan^2 x + \sec^2 x = \tan^2 x + (1 + \tan^2 x)

Simplifying:

tan⁡2x+1+tan⁡2x=1+2tan⁡2x\tan^2 x + 1 + \tan^2 x = 1 + 2 \tan^2 x

This confirms that the identity is true. Now we examine the four options to determine which one demonstrates this correctly.

• Option A rewrites sec²x as 1−tan⁡2×1 – \tan^2 x, which is incorrect. There is no valid identity where sec²x equals 1 minus tan²x.
• Option B rewrites sec²x as 1+tan⁡2×1 + \tan^2 x, which is accurate. It substitutes this correctly and simplifies it to 1+2tan⁡2×1 + 2 \tan^2 x.
• Option C uses sec⁡2x=tan⁡2x−1\sec^2 x = \tan^2 x – 1, which is also incorrect and violates the Pythagorean identity.
• Option D states that tan⁡2x+sec⁡2x=1\tan^2 x + \sec^2 x = 1, which contradicts the original identity and does not follow logically.

Therefore, Statement B is correct because it applies the identity sec⁡2x=1+tan⁡2x\sec^2 x = 1 + \tan^2 x and correctly shows how the expression simplifies to match the right-hand side.

This identity is a good example of how recognizing fundamental trigonometric identities can help simplify more complex expressions. It emphasizes the importance of substitution and step-by-step simplification in proving identities.