Answer
$\theta = 10^{\circ}$
Work Step by Step
Use the concept of cofunction $\sec x = \csc (90^{\circ} -x)$
$\sec (2\theta +10^{\circ}) = \csc (4\theta +20^{\circ})$
$\sec (2\theta +10^{\circ}) = \sec (90^{\circ} - (4\theta +20^{\circ}))$
$2\theta +10 ^{\circ} = 90^{\circ} -4\theta -20^{\circ}$
$6\theta =60^{\circ}$
$\theta =10^{\circ}$
