$A$ and $B$ are positive acute angles such that $\cos 2B=3{\sin }^{2}A$ and $3\sin 2A=2\sin 2B$. What is the value of $\left(A+2B\right)$?

If $\sin \theta =\frac{1}{\sqrt{2}}$, then $(\tan \theta + \cos \theta )$ = 

1. $\frac{1}{\sqrt{2}}$
2. $\frac{2}{\sqrt{2}}$ 
3. $\frac{3}{\sqrt{2}}$
4. $\frac{(1+\sqrt{2})}{\sqrt{2}}$

If $\cos \theta =\frac{2p}{p^2+1}$, $\left(p\ne \pm 1\right)$ then $\mathrm{cosec}\theta$ is equal to:

The value of $(\sin \theta + \cos \theta {)}^2=?$

1. $1+{\sin }^2\theta$
2. ${\sin }^2\theta +{\cos }^2\theta$ 
3.$1+2 \cos \theta \sin \theta$ 
4. ${\cos }^2 \theta + 1$

What is the value of ${\cos }^{2}5°+{\cos }^{2}10°+ \_\_\_\_\_+{\cos }^{2}90°$

What is the simplified value of: $\frac{{\sin }^3 {21}^{\circ }+{\cos }^3 {19}^{\circ }}{\sin {21}^{\circ }+\cos {19}^{\circ }}+{\sin }^2 {69}^{\circ }+{\cos }^2 {71}^{\circ }+\frac{1}{\sec {69}^{\circ }\mathrm{cosec} {71}^{\circ }}$ is

If $L={\sin }^2\left(\frac{\pi }{16}\right)-{\sin }^2\left(\frac{\pi }{8}\right)$ and $M={\cos }^2\left(\frac{\pi }{16}\right)-{\sin }^2\left(\frac{\pi }{8}\right)$