[tex]\bf \textit{symmetry identities}\\\\
sin(-\theta )\implies -sin(\theta )\qquad \qquad cos(-\theta )\implies cos(\theta )
\\\\\\also~recall\\\\ sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta)
\\\\\\
sin^2(\theta)=1-cos^2(\theta)
\\\\
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[1-cos(-t)][1+cos(t)]\implies [1-cos(t)][1+cos(t)][/tex]
[tex]\bf 1^2-cos^2(t)\implies 1-cos^2(t)\implies sin^2(t)\\\\
-------------------------------\\\\\
%Simplify each expression. (1−cos(−t))(1+cos(t)) = (1+sin(t))(1+sin(-t))= csc(t)tan(t)+sec(−t) =
[1+sin(t)][1+sin(-t)]\implies [1+sin(t)][1-sin(t)]
\\\\\\
1^2-sin^2(t)\implies 1-sin^2(t)\implies cos^2(t)\\\\
-------------------------------\\\\[/tex]
[tex]\bf csc(t)tan(t)+sec(-t)\implies \cfrac{1}{\underline{sin(t)}}\cdot \cfrac{\underline{sin(t)}}{cos(t)}+\cfrac{1}{cos(-t)}
\\\\\\
\cfrac{1}{cos(t)}+\cfrac{1}{cos(-t)}\implies \cfrac{1}{cos(t)}+\cfrac{1}{cos(t)}\implies \cfrac{2}{cos(t)}
\\\\\\
2\cdot \cfrac{1}{cos(t)}\implies 2sec(t)[/tex]
