Verify the Identity tan(2x)^2+sin(2x)^2+cos(2x)^2=sec(2x)^2

Problem

tan(2*x)+sin(2*x)+cos(2*x)=sec(2*x)

Solution

1. Identify the Pythagorean identity for sine and cosine, which states that sin(θ)+cos(θ)=1

2. Substitute the identity into the left side of the equation by replacing sin(2*x)+cos(2*x) with 1

tan(2*x)+1=sec(2*x)

3. Apply the Pythagorean identity for tangent and secant, which states that tan(θ)+1=sec(θ)

4. Conclude that the left side of the equation is now identical to the right side.

sec(2*x)=sec(2*x)

Final Answer

tan(2*x)+sin(2*x)+cos(2*x)=sec(2*x)

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