[tex]\bf sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta)
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sec(\theta)=\cfrac{1}{cos(\theta)}\\\\
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\cfrac{1}{1+sin(\theta)}+\cfrac{1}{1-sin(\theta)}\impliedby LCD=[1-sin(\theta)][1+sin(\theta)]
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\cfrac{1\underline{-sin(\theta)}+1\underline{+sin(\theta)}}{[1-sin(\theta)][1+sin(\theta)]}\implies
\cfrac{2}{[1-sin(\theta)][1+sin(\theta)]}\\\\
-------------------------------\\\\[/tex]
[tex]\bf \textit{difference of squares}
\\ \quad \\
(a-b)(a+b) = a^2-b^2\qquad \qquad
a^2-b^2 = (a-b)(a+b)\\\\
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\cfrac{2}{1^2-sin^2(\theta)}\implies \cfrac{2}{1-sin^2(\theta)}\implies \cfrac{2}{cos^2(\theta)}
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\cfrac{2}{1}\cdot \cfrac{1^2}{cos^2(\theta)}\implies 2sec^2(\theta)[/tex]
