Prove that, if \(\tan\theta\tan(\theta+\alpha)=k\), then \[(k+1)\cos(2\theta+\alpha)=(1-k)\cos\alpha.\]

Solve the equation \[\tan\theta\tan(\theta+\pi/3)=2.\]

Discuss the equation \[\tan\theta\tan(\theta+\alpha)+1=0.\]

Prove that, if \(\tan\theta\tan(\theta+\alpha)=k\), then \[(k+1)\cos(2\theta+\alpha)=(1-k)\cos\alpha.\]

Solve the equation \[\tan\theta\tan(\theta+\pi/3)=2.\]

Discuss the equation \[\tan\theta\tan(\theta+\alpha)+1=0.\]