$(1-\cos \theta)(1+\cos \theta) = \dots$

Given that $\sin \theta + 2 \cos \theta = 1$,prove that $2 \sin \theta - \cos \theta = 2$.

If $\sec ^{2} \theta+\tan ^{2} \theta=\frac{13}{12}$,then the value of $\sec ^{4} \theta-\tan ^{4} \theta$ is .........

Prove that,$(\sin \alpha+\cos \alpha)(\tan \alpha+\cot \alpha)=\sec \alpha+\operatorname{cosec} \alpha$

If $\tan 7 \theta \cdot \tan 3 \theta = 1,$ then the value of $\theta$ is ..........

Prove the following identity,where the angles involved are acute angles for which the expressions are defined:
$\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta$