Sine Plus Cosine times Tangent Plus Cotangent
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Theorem
- $\paren {\sin x + \cos x} \paren {\tan x + \cot x} = \sec x + \csc x$
Proof
| \(\ds \paren {\sin x + \cos x} \paren {\tan x + \cot x}\) | \(=\) | \(\ds \paren {\sin x + \cos x} \paren {\sec x \csc x}\) | Sum of Tangent and Cotangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {\sin x + \cos x} {\sin x \cos x}\) | Definition of Secant Function and Definition of Cosecant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {\cos x} + \frac 1 {\sin x}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sec x + \csc x\) | Definition of Secant Function and Definition of Cosecant |
$\blacksquare$