4.02: Trigonometric identities and equations

Show that \(\tan\theta+\cot\theta+2\sec\theta\) can be written as \(\dfrac{1+2\sin\theta}{\sin\theta\cos\theta}\).

Hence solve the equation \(\tan2x+\cot2x+2\sec2x=0\) for \(-180^\circ\leqslant x\leqslant180^\circ\).

Solve the equation \(3\sec(x-1)-1=\tan^2(x-1)\), where \(x\) is in radians and \(-1<x<3\).