Trigonometric Value Calculation

Given the equation $\sin A + \cos A = 1$, we need to find the value of $\sin^4 A + \cos^4 A$.

Step-by-Step Solution

1. Square the given equation:

   Starting with $\sin A + \cos A = 1$. Squaring both sides yields:

   $(\sin A + \cos A)^2 = 1^2$

   Expanding this gives: $\sin^2 A + \cos^2 A + 2 \sin A \cos A = 1$

2. Apply the Pythagorean identity:

   Using the fundamental identity $\sin^2 A + \cos^2 A = 1$, the equation becomes:

   $1 + 2 \sin A \cos A = 1$

3. Solve for $\sin A \cos A$:

   Subtracting 1 from both sides simplifies the equation to $2 \sin A \cos A = 0$.

   Therefore, $\sin A \cos A = 0$.

4. Determine possible values for $\sin A$ and $\cos A$:

   The condition $\sin A \cos A = 0$ implies either $\sin A = 0$ or $\cos A = 0$.

   • If $\sin A = 0$, the original equation $\sin A + \cos A = 1$ implies $\cos A = 1$.
   • If $\cos A = 0$, the original equation $\sin A + \cos A = 1$ implies $\sin A = 1$.

5. Calculate the target expression $\sin^4 A + \cos^4 A$:

   We check both possibilities:

   • Case 1: $\sin A = 0, \cos A = 1$.

     Then $\sin^4 A + \cos^4 A = 0^4 + 1^4 = 0 + 1 = 1$.

   • Case 2: $\sin A = 1, \cos A = 0$.

     Then $\sin^4 A + \cos^4 A = 1^4 + 0^4 = 1 + 0 = 1$.

Thus, the value of $\sin^4 A + \cos^4 A$ is 1.