Prove that: $\sin x+\sin 3 x+\sin 5 x+\sin 7 x=4 \cos x \cos 2 x \sin 4 x$
It is known that $\sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cdot \cos \left(\frac{A-B}{2}\right)$
$L.H.S.$ $=\sin x+\sin 3 x+\sin 5 x+\sin 7 x$
$=(\sin x+\sin 5 x)+(\sin 3 x+\sin 7 x)$
$=2 \sin \left(\frac{x+5 x}{2}\right) \cdot \cos \left(\frac{x-5 x}{2}\right)+2 \sin \left(\frac{3 x+7 x}{2}\right) \cos \left(\frac{3 x-7 x}{2}\right)$
$=2 \sin 3 x \cos (-2 x)+2 \sin 5 x \cos (-2 x)$
$=2 \sin 3 x \cos 2 x+2 \sin 5 x \cos 2 x$
$=2 \cos 2 x[\sin 3 x+\sin 5 x]$
$=2 \cos 2 x\left[2 \sin \left(\frac{3 x+5 x}{2}\right) \cdot \cos \left(\frac{3 x-5 x}{2}\right)\right]$
$=2 \cos 2 x[2 \sin 4 x \cdot \cos (-x)]$
$=4 \cos 2 x \sin 4 x \cos x=R . H . S.$
Prove that: $\frac{(\sin 7 x+\sin 5 x)+(\sin 9 x+\sin 3 x)}{(\cos 7 x+\cos 5 x)+(\cos 9 x+\cos 3 x)}=\tan 6 x$
If $\tan \theta = \frac{{20}}{{21}},$ cos$\theta$ will be
Prove that $\cos \left(\frac{\pi}{4}-x\right) \cos \left(\frac{\pi}{4}-y\right)-\sin \left(\frac{\pi}{4}-x\right) \sin \left(\frac{\pi}{4}-y\right)=\sin (x+y)$
If $\tan \theta - \cot \theta = a$ and $\sin \theta + \cos \theta = b,$ then ${({b^2} - 1)^2}({a^2} + 4)$ is equal to
Find the radian measures corresponding to the following degree measures:
$-47^{\circ} 30^{\prime}$