Difference of Fourth Powers of Secant and Tangent
From ProofWiki
Theorem
- $\sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x$
Proof
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \sec^4 x - \tan^4 x\) | \(=\) | \(\displaystyle \frac 1 {\cos^4 x} - \frac {\sin^4 x} {\cos^4 x}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | by definition of secant and tangent | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {1 - \sin^4 x} {\cos^4x}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {1 - \sin^2 x \left({1 - \cos^2 x}\right)} {\cos^2 x \left({1 - \sin^2 x}\right)}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Sum of Squares of Sine and Cosine | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {1 - \sin^2 x + \sin^2 x \ \cos^2 x} {\cos^2 x - \sin^2 x \ \cos^2 x}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\cos^2 x + \sin^2 x \ \cos^2 x} {\cos^2 x - \sin^2 x \ \cos^2 x}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Sum of Squares of Sine and Cosine | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {\cos^2 x \left({1 + \sin^2 x}\right)} {\cos^2 x \left({1 - \sin^2 x}\right)}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {1 + \sin^2 x} {1 - \sin^2 x}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac {1 + \sin^2 x} {\cos^2 x}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | Sum of Squares of Sine and Cosine | ||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \frac 1 {\cos^2 x} + \frac {\sin^2 x} {\cos^2 x}\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(=\) | \(\displaystyle \sec^2 x + \tan^2 x\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | by definition of secant and tangent |
$\blacksquare$
