Verify the Identity cos(2x)^2-sin(2x)^2=0

Problem

cos(2*x)−sin(2*x)=0

Solution

1. Identify the expression on the left side of the equation as a specific form of a trigonometric identity.

2. Apply the double-angle formula for cosine, which states cos(2*θ)=cos(θ)−sin(θ)

3. Substitute θ=2*x into the identity to rewrite the expression.

cos(2*(2*x))=cos(2*x)−sin(2*x)

4. Simplify the argument of the cosine function.

cos(4*x)=cos(2*x)−sin(2*x)

5. Compare the simplified expression to the right side of the original equation.

cos(4*x)=0

6. Conclude that the statement cos(2*x)−sin(2*x)=0 is a conditional equation that is true only for specific values of x where cos(4*x)=0 rather than an identity that is true for all x

Final Answer

cos(2*x)−sin(2*x)=cos(4*x)

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