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:
$\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\operatorname{cosec} A+\cot A$,using the identity $\operatorname{cosec}^{2} A=1+\cot ^{2} A$.

If $\tan \theta + \sec \theta = l$,then prove that $\sec \theta = \frac{l^{2} + 1}{2l}$.

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

Prove that $\sec A(1-\sin A)(\sec A+\tan A)=1$.

Prove that $\frac{\sin \theta-\cos \theta+1}{\sin \theta+\cos \theta-1}=\frac{1}{\sec \theta-\tan \theta},$ using the identity $\sec ^{2} \theta=1+\tan ^{2} \theta.$