Use the cofunction identities to find an angle $\theta$ between $0^{\circ}$ and $90^{\circ}$ that makes the statement frue. $\sec \theta=\csc \left(4 \theta+60^{\circ}\right)$
$\theta=\square^{\circ}$ (Type an integer or a simplified fraction.)

Step 1 ：The cofunction identities in trigonometry state that the secant of an angle is equal to the cosecant of the complement of that angle. In other words, \(\sec(\theta) = \csc(90^\circ - \theta)\).

Step 2 ：Given the equation \(\sec(\theta) = \csc(4\theta + 60^\circ)\), we can use the cofunction identity to rewrite the right side of the equation as \(\sec(90^\circ - (4\theta + 60^\circ))\).

Step 3 ：This gives us the equation \(\sec(\theta) = \sec(30^\circ - 4\theta)\).

Step 4 ：We can solve this equation for \(\theta\) by setting the two sides equal to each other and solving for \(\theta\).

Step 5 ：The solution to the equation is \(\theta = 6\). This means that the angle \(\theta\) that makes the statement \(\sec \theta=\csc \left(4 \theta+60^\circ\right)\) true is \(6^\circ\).

Step 6 ：Final Answer: \(\boxed{6^\circ}\)