Sum of Squares of Sine and Cosine/Proof 1
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Theorem
- $\cos^2 x + \sin^2 x = 1$
Proof
\(\ds 1\) | \(=\) | \(\ds \cos 0\) | Cosine of Zero is One | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \cos {x - x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos x \map \cos {-x} - \sin x \map \sin {-x}\) | Cosine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \cos x \cos x - \paren {-\sin x \sin x}\) | Cosine Function is Even and Sine Function is Odd | |||||||||||
\(\ds \) | \(=\) | \(\ds \cos^2 x + \sin^2 x\) |
$\blacksquare$
Warning
Note that we need to start from the algebraic definitions of sine and cosine:
- $\ds \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$
- $\ds \cos x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \cdots$
and then use the proofs of the Cosine of Sum that derive directly from these.
Otherwise these proofs are circular.
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $(4.20)$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 16.3 \ (3) \ \text{(i)}$