Question

Use fundamental identities to fully simply the expression.

Answer 1

$\bf \stackrel{\textit{pythagorean identities}}{sin^2(\theta)+cos^2(\theta)=1}\qquad \qquad \stackrel{\textit{symmetry identities}}{cos(-\theta)=cos(\theta ) } \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \begin{array}{llrll} 3sin^3(t)csc(t)\qquad &+cos^2(t)&\qquad +2cos(-t)cos(t)\\\\ 3sin^3(t)\cdot \cfrac{1}{sin(t)}&&+2[cos(t)]cos(t)\\\\ 3sin^2(t)&&+2cos^2(t) \end{array} \\\\[-0.35em] ~\dotfill\\\\ 3sin^2(t)+cos^2(t)+2cos^2(t)\implies 3sin^2(t)+3cos^2(t) \\\\\\ 3[sin^2(t)+cos^2(t)]\implies 3[1]\implies 3$