prove this identity $\sin(x+y)\sin(xy)=\sin^2 x  \sin^2 y$
12,981
Solution 1
$\sin^2 x \cos^2 y\cos^2 x \sin^2 y=\sin^2 x(1\sin^2 y) (1\sin^2 x) \sin ^2 y$
$=\sin^2 x \sin^2 x\sin^2y \sin ^2 y + \sin^2x\sin^2y$
$=\sin^2 x  \sin ^2 y$
Solution 2
Use the identity $$\sin(x\pm y)=\sin x\cos y\pm \sin y\cos x$$ and we can get \begin{align*} LHS&=\sin(x+y)\sin(xy)\\ &=(\sin x\cos y+\cos x\sin y)(\sin x\cos y\cos x\sin y)\\ &=\sin^2x\cos^2y\cos^2x\sin^2y \\ \end{align*}
See if you can take it from here using the identity $$\sin^2x+\cos^2x=1.$$
Related videos on Youtube
Author by
dona12
Updated on July 21, 2022Comments

dona12 less than a minute
prove this identity : $$\sin(x+y)\sin(xy)=\sin^2 x  \sin^2 y$$ I tried solving it with additional formulas but I can't get the right answer. I get $$\sin^2 x \cos^2 y\cos^2 x \sin^2 y$$

Ragnar over 8 yearsDo you want a geometrical proof or just a proof using some common facts like $\sin^2+\cos^2=1$?

dona12 over 8 yearsI think common facts

MartínBlas Pérez Pinilla over 8 yearsDo you know the formulas for $\sin(x\pm y)$?
