Proof of the Identity
To prove the identity:
cos²A + sin²A = sinA + cos²A
we can use the Pythagorean identity and algebraic manipulation.
Step 1: Start with the Pythagorean Identity
- Recall the fundamental trigonometric identity:
cos²A + sin²A = 1
Step 2: Substitute the Pythagorean Identity
- Replace cos²A + sin²A in the original equation:
1 = sinA + cos²A
Step 3: Rearrange the Equation
- Rearranging gives us:
cos²A = 1 - sinA
Step 4: Verify the Rearranged Equation
- We can check if the equation holds true by substituting back:
Substitute cos²A into the identity:
1 - sinA + sin²A = sinA + (1 - sinA)
This simplifies to:
1 - sinA + sin²A = 1
Conclusion
- Since both sides of the equation are equal, we have verified that:
cos²A + sin²A = sinA + cos²A
Thus, the identity holds true.
Key Takeaway
- The proof relies on the application of the Pythagorean identity and basic algebraic manipulation, confirming the equality of both sides of the equation.