Product of One Plus Cotangent with One Plus Tangent
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Theorem
- $\paren {1 + \cot x} \paren {1 + \tan x} = 2 + \csc x \sec x$
Proof
| \(\ds \paren {1 + \cot x} \paren {1 + \tan x}\) | \(=\) | \(\ds 1 + \cot x + \tan x + \cot x \tan x\) | multiplying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 + \cot x + \tan x + \dfrac 1 {\tan x} \tan x\) | Cotangent is Reciprocal of Tangent | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 + \cot x + \tan x\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 + \csc x \sec x\) | Sum of Tangent and Cotangent |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Exercise $\text {XXXI}$: $2.$