Problem 39 - Precalculus Essentials

Question

Verify each identity. $$\tan ^{2} 2 x+\sin ^{2} 2 x+\cos ^{2} 2 x=\sec ^{2} 2 x$$

Step-by-Step Solution

Answer

By applying the Pythagorean identity and simplifying the equation, we see both sides of the original identity match, verifying that \(tan^22x + sin^22x + cos^22x = sec^22x\).

Step 1: Transform terms using identities

The problem can initially be simplified by transforming the secant term using the identity \(sec^2x = 1 + tan^2x\). Now the equation becomes: \(tan^2(2x) + sin^2(2x) + cos^2(2x) = 1 + tan^2(2x)\).

Step 2: Simplify equation

Now the equation can be further simplified by subtracting \(tan^22x\) from both sides and rearranging terms: \(sin^22x + cos^22x = 1\).

Step 3: Apply the Pythagorean identity

The left hand side of the equation is \( sin^22x + cos^22x \), which is recognized as one of the fundamental Pythagorean identities in trigonometry. This equals to 1, hence both sides of the equality match, and the original identity is verified.