The value of ${e^{{{\log }_{10}}\tan 1^\circ + {{\log }_{10}}\tan 2^\circ + {{\log }_{10}}\tan 3^\circ + ........... + {{\log }_{10}}\tan 89^\circ }}$ is

$\text{cosec } A - 2 \cot 2A \cos A = $

If $2 \sin \theta + \cos \theta = \frac{7}{3},$ then the value of $(\tan^2 \theta - \sec^2 \theta)$ is

The value of $\sin ^{2} 75^{\circ}-\sin ^{2} 15^{\circ}$ is:

If $\sin A = n \sin B,$ then $\frac{n - 1}{n + 1} \tan \frac{A + B}{2} = $

Consider the following two statements.
Statement $p$: The value of $\sin 120^\circ$ can be derived by taking $\theta = 240^\circ$ in the equation $2 \sin \frac{\theta}{2} = \sqrt{1 + \sin \theta} - \sqrt{1 - \sin \theta}$.
Statement $q$: The angles $A, B, C$ and $D$ of any quadrilateral $ABCD$ satisfy the equation $\cos \left( \frac{1}{2}(A + C) \right) + \cos \left( \frac{1}{2}(B + D) \right) = 0$.
Then the truth values of $p$ and $q$ are respectively: