Sine Plus Cosine times Tangent Plus Cotangent

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Theorem

$\left({\sin x + \cos x}\right) \left({\tan x + \cot x}\right) = \sec x + \csc x$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \left({\sin x + \cos x}\right) \left({\tan x + \cot x}\right)$	$=$	$\displaystyle \left({\sin x + \cos x}\right) \left({\sec x \ \csc x}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Sum of Tangent and Cotangent
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac{\sin x + \cos x} {\sin x \ \cos x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of secant and cosecant
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac 1 {\cos x} + \frac 1 {\sin x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \sec x + \csc x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	by definition of secant and cosecant

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({\sin x + \cos x}\right) \left({\tan x + \cot x}\right)\)	\(=\)	\(\displaystyle \left({\sin x + \cos x}\right) \left({\sec x \ \csc x}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Sum of Tangent and Cotangent
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac{\sin x + \cos x} {\sin x \ \cos x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of secant and cosecant
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac 1 {\cos x} + \frac 1 {\sin x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sec x + \csc x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of secant and cosecant