If cos A + cos²A = 1, then what is the value of sin²A + sin⁴A?

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Given: cos A + cos²A = 1

Step 1: Let x = cos A.
Then the equation becomes:

x + x² = 1
=> x² + x - 1 = 0

Step 2: We want to find sin²A + sin⁴A.
Recall that sin²A = 1 - cos²A = 1 - x².

Step 3: Calculate sin²A + sin⁴A:
= sin²A + (sin²A)²
= y + y², where y = sin²A = 1 - x²

So,
y + y² = (1 - x²) + (1 - x²)²
= (1 - x²) + (1 - 2x² + x^4)
= 1 - x² + 1 - 2x² + x^4
= 2 - 3x² + x^4

Step 4: Use the quadratic equation for x:
x² + x - 1 = 0
=> x² = 1 - x

Step 5: Substitute x² = 1 - x into the expression:

2 - 3x² + x^4 = 2 - 3(1 - x) + (x²)²
= 2 - 3 + 3x + (x²)²
= -1 + 3x + (x²)²

Step 6: Calculate (x²)² = x^4
Since x² = 1 - x, then x^4 = (x²)² = (1 - x)² = 1 - 2x + x²

Substitute back:
-1 + 3x + 1 - 2x + x² = (-1 + 1) + (3x - 2x) + x² = 0 + x + x²

Step 7: Again substitute x² = 1 - x:

x + x² = x + (1 - x) = 1

Therefore, sin²A + sin⁴A = 1.

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