$\cos (40^{\circ}-\theta)-\sin (50^{\circ}+\theta) = \ldots \ldots \ldots \ldots$

$\sin (45^{\circ}+\theta)-\cos (45^{\circ}-\theta)$ is equal to

Show that $\frac{\cos ^{2}\left(45^{\circ}+\theta\right)+\cos ^{2}\left(45^{\circ}-\theta\right)}{\tan \left(60^{\circ}+\theta\right) \tan \left(30^{\circ}-\theta\right)}=1$

Prove the following identity,where the angles involved are acute angles for which the expressions are defined:
$\frac{1+\sec A}{\sec A} = \frac{\sin^2 A}{1-\cos A}$

State whether the following is 'True' or 'False' and justify your answer:
If $\cos A + \cos^2 A = 1$,then $\sin^2 A + \sin^4 A = 1$.

If $\operatorname{cosec} \theta + \cot \theta = p$,then prove that $\cos \theta = \frac{p^{2} - 1}{p^{2} + 1}$.