Answer: \(\theta=39.2^\circ,\ 140.8^\circ\).

Use the standard trigonometric identities and make sure the final angles are chosen from the interval given in the question.

Expand the left-hand side:

\((2\tan\theta+\sec\theta)(2\tan\theta-\sec\theta)=4\tan^2\theta-\sec^2\theta.\)

Using \(\sec^2\theta=1+\tan^2\theta\),

\(4\tan^2\theta-\sec^2\theta=4\tan^2\theta-(1+\tan^2\theta)=3\tan^2\theta-1.\)

For part (b), use the result from part (a):

\(3\tan^2\theta-1=1.\)

So \(3\tan^2\theta=2\), and hence \(\tan\theta=\pm\sqrt{\frac{2}{3}}\).

In the interval \(0^\circ\leqslant\theta\leqslant180^\circ\), this gives

\(\theta=39.2^\circ\) or \(\theta=140.8^\circ\).

This gives the required answer: \(\theta=39.2^\circ,\ 140.8^\circ\).