Answer
We want to verify that $sec \ t\ csc\ t (tan\ t +cot\ t)=sec^2t+csc^2t$
We start with the right-hand side:
$sec \ t\ csc\ t (tan\ t +cot\ t)$
$=\frac{1}{cos\ t}\frac{1}{sin\ t} (\frac{sin\ t}{cos\ t} +\frac{cos\ t}{sin\ t})$
$=\frac{sin\ t}{cos^2\ t\ sin\ t}+\frac{cos\ t}{cos\ t\ sin^2\ t}$
$=\frac{1}{cos^2\ t}+\frac{1}{sin^2\ t}$
$=sec^2t+csc^2t$, which is the left-hand side, thus verifying the identity.
Work Step by Step
We want to verify that $sec \ t\ csc\ t (tan\ t +cot\ t)=sec^2t+csc^2t$
We start with the right-hand side:
$sec \ t\ csc\ t (tan\ t +cot\ t)$
$=\frac{1}{cos\ t}\frac{1}{sin\ t} (\frac{sin\ t}{cos\ t} +\frac{cos\ t}{sin\ t})$
$=\frac{sin\ t}{cos^2\ t\ sin\ t}+\frac{cos\ t}{cos\ t\ sin^2\ t}$
$=\frac{1}{cos^2\ t}+\frac{1}{sin^2\ t}$
$=sec^2t+csc^2t$, which is the left-hand side, thus verifying the identity.
