If $\sin \theta = \sin 15^{\circ} + \sin 45^{\circ}$,where $0^{\circ} < \theta < 180^{\circ}$,then $\theta =$ (in $^{\circ}$)

Prove that: $(\cos x+\cos y)^{2}+(\sin x-\sin y)^{2}=4 \cos ^{2} \frac{x+y}{2}$

The value of $\cos(18^{\circ}-A) \cos(18^{\circ}+A) - \cos(72^{\circ}-A) \cos(72^{\circ}+A)$ is equal to

If $(1 + \tan \theta )(1 + \tan \phi ) = 2$,then $\theta + \phi = \dots ^\circ$

Let $\cos(\alpha+\beta)=-\frac{1}{10}$ and $\sin(\alpha-\beta)=\frac{3}{8}$ where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha=\frac{3(1-r\sqrt{5})}{\sqrt{11}(s+\sqrt{5})}$,where $r, s \in N$,then $r+s$ is equal to . . . . . .

If $\frac{x}{\cos \alpha} = \frac{y}{\cos \left(\frac{2 \pi}{3} - \alpha\right)} = \frac{z}{\cos \left(\frac{2 \pi}{3} + \alpha\right)}$,then the value of $(x + y + z)$ is equal to