Verify the Identity sin(x)^2=4-2cos(x)^2

Problem

sin2(x)=4−2*cos2(x)

Solution

1. Identify the goal, which is to determine if the given equation sin2(x)=4−2*cos2(x) is a valid trigonometric identity for all values of x

2. Apply the Pythagorean identity sin2(x)=1−cos2(x) to rewrite the left side of the equation in terms of cos(x)

3. Substitute the identity into the original equation to compare both sides.

1−cos2(x)=4−2*cos2(x)

4. Rearrange the equation by adding 2*cos2(x) to both sides to see if it results in a statement that is always true.

1+cos2(x)=4

5. Solve for cos2(x) to find the specific values where this equation holds.

cos2(x)=3

6. Evaluate the result. Since the range of cos(x) is [−1,1] the maximum value of cos2(x) is 1 Therefore, cos2(x)=3 has no real solutions.

Final Answer

sin2(x)=4−2*cos2(x)* is not an identity.

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