Solve for x cos(x)^2-sin(x)^2=1-2sin(x)^2

Problem

cos(x)−sin(x)=1−2*sin(x)

Solution

1. Identify the trigonometric identity for cos(x) Recall that cos(x)=1−sin(x)

2. Substitute this identity into the left side of the equation.

cos(x)−sin(x)=(1−sin(x))−sin(x)

3. Simplify the expression by combining the like terms on the left side.

1−sin(x)−sin(x)=1−2*sin(x)

4. Compare both sides of the equation. Since the left side simplifies to 1−2*sin(x) the equation becomes an identity.

1−2*sin(x)=1−2*sin(x)

5. Conclude that because the equation is true for all values of x where the functions are defined, the solution is all real numbers.

Final Answer

cos(x)−sin(x)=1−2*sin(x)⇒x∈ℝ

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