If $\sin \theta + \cos \theta = 1$, then $\sin \theta \cos \theta = $

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

Find the angle in radian through which a pendulum swings if its length is $75\, cm$ and the tip describes an arc of length.

$15\,cm$

Find $\sin \frac{x}{2}, \cos \frac{x}{2}$ and $\tan \frac{x}{2}$ for $\sin x=\frac{1}{4}, x$ in quadrant $II$

Find the radian measures corresponding to the following degree measures:

$-47^{\circ} 30^{\prime}$

If $\tan \theta - \cot \theta = a$ and $\sin \theta + \cos \theta = b,$ then ${({b^2} - 1)^2}({a^2} + 4)$ is equal to