Question

Trigonometric Identities

Simplify each expression.
(1−cos⁡(−t))(1+cos⁡(t)) =
(1+sin(t))(1+sin(-t))=
csc⁡(t)tan⁡(t)+sec⁡(−t) =
Thank you for your help

Answer 1

`\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) \\\\ -------------------------------\\\\\ [1-cos(-t)][1+cos(t)]\implies [1-cos(t)][1+cos(t)]`


`\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)\\\\ -------------------------------\\\\`


`\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)`