Answer
$\csc(-56^\circ42')$ is the cofunction needed to be found.
Work Step by Step
$$\sec146^\circ42'$$
Cosecant and secant are cofunctions. So to write $\sec146^\circ42'$ in terms of cofunction, cosecant would be included.
Now the job is to figure out the value of $\theta$, which must satisfy
$$\csc\theta=\sec146^\circ42'\hspace{1cm}(1)$$
According to Cofunction Identity: $\csc\theta=\sec(90^\circ-\theta)$
Apply this to the equation $(1)$:
$$\sec(90^\circ-\theta)=\sec146^\circ42'$$
$$90^\circ-\theta=146^\circ42'$$
$$\theta=90^\circ-146^\circ42'=-56^\circ42'$$
$\csc(-56^\circ42')$ is the answer to this exercise.
