Why is sin(x) = 1 - cos(x) not an identity?

Steps to Solve

1. Evaluate Both Sides of the Equation
   Start by examining values for $x$ and calculating both sides of the equation $sin(x) = 1 - cos(x)$ to see if they are equal.

2. Test Specific Angles
   Choose specific values for $x$, like $0$, $\frac{\pi}{2}$, and $\pi$, and evaluate both sides.

• For $x = 0$:

  • Left side: $sin(0) = 0$
  • Right side: $1 - cos(0) = 1 - 1 = 0$ (both sides are equal)

• For $x = \frac{\pi}{2}$:

  • Left side: $sin\left(\frac{\pi}{2}\right) = 1$
  • Right side: $1 - cos\left(\frac{\pi}{2}\right) = 1 - 0 = 1$ (both sides are equal)

• For $x = \pi$:

  • Left side: $sin(\pi) = 0$
  • Right side: $1 - cos(\pi) = 1 - (-1) = 2$ (both sides are not equal)

3. Check General Cases
   Investigate more angle values generally to see if both sides hold for all angles. Notice that they do not equal for all values, especially when $x$ is not among those tested.

4. Conclusion on Non-Identity
   Since there are angles where the left side does not equal the right side, $sin(x) = 1 - cos(x)$ cannot be a trigonometric identity. An identity must hold true for all values of $x$.