Difference of Fourth Powers of Secant and Tangent
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Theorem
- $\sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x$
Proof
| \(\ds \sec^4 x - \tan^4 x\) | \(=\) | \(\ds \sec^4 x - \frac {\sin^4 x} {\cos^4 x}\) | Tangent is Sine divided by Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {\cos^4 x} - \frac {\sin^4 x} {\cos^4 x}\) | Secant is Reciprocal of Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 - \sin^4 x} {\cos^4x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 - \sin^2 x \paren {1 - \cos^2 x} } {\cos^2 x \paren {1 - \sin^2 x} }\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 - \sin^2 x + \sin^2 x \ \cos^2 x} {\cos^2 x - \sin^2 x \ \cos^2 x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\cos^2 x + \sin^2 x \ \cos^2 x} {\cos^2 x - \sin^2 x \ \cos^2 x}\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\cos^2 x \paren {1 + \sin^2 x} } {\cos^2 x \paren {1 - \sin^2 x} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 + \sin^2 x} {1 - \sin^2 x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {1 + \sin^2 x} {\cos^2 x}\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {\cos^2 x} + \frac {\sin^2 x} {\cos^2 x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {\cos^2 x} x + \tan^2 x\) | Tangent is Sine divided by Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sec^2 x + \tan^2 x\) | Secant is Reciprocal of Cosine |
$\blacksquare$