Sine Plus Cosine times Tangent Plus Cotangent

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

$\blacksquare$

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