If $A + B + C = 270^o,$ then $\cos 2A + \cos 2B + \cos 2C + 4\sin A \sin B \sin C = $

$\cos A + \sin (270^\circ + A) - \sin (270^\circ - A) + \cos (180^\circ + A) = $

If $\cos A = \frac{3}{4}$,then $32\sin \left( \frac{A}{2} \right)\sin \left( \frac{5A}{2} \right) = $

The maximum value of $(7 \cos\theta + 24 \sin\theta) \times (7 \sin\theta - 24 \cos\theta)$ for every $\theta \in R$.

The value of $\cos (270^\circ + \theta )\,\cos (90^\circ - \theta ) - \sin (270^\circ - \theta )\,\cos \theta $ is

If $a \cos^3 \alpha + 3a \cos \alpha \sin^2 \alpha = m$ and $a \sin^3 \alpha + 3a \cos^2 \alpha \sin \alpha = n$,then $(m + n)^{2/3} + (m - n)^{2/3}$ is equal to