Simplify the expression algebraically.
$\sin ( 9 x + 9 y ) \sin ( 9 x - 9 y )$
A) $\sin ^ { 2 } x - \sin ^ { 2 } 9 y$
B) $\sin ^ { 2 } 9 x + \sin ^ { 2 } 9 y$
C) $\sin ^ { 2 } 9 x - \sin ^ { 2 } 9 y$
D) $\sin ^ { 2 } x + \sin ^ { 2 } y$
E) $\sin ^ { 2 } 9 x - \sin ^ { 2 } y$

Question

Simplify the expression algebraically. 
 $\sin ( 9 x + 9 y ) \sin ( 9 x - 9 y )$  
A) $\sin ^ { 2 } x - \sin ^ { 2 } 9 y$
B) $\sin ^ { 2 } 9 x + \sin ^ { 2 } 9 y$
C) $\sin ^ { 2 } 9 x - \sin ^ { 2 } 9 y$
D) $\sin ^ { 2 } x + \sin ^ { 2 } y$
E) $\sin ^ { 2 } 9 x - \sin ^ { 2 } y$

Quizplus · Accepted Answer

This expression can be simplified using the trigonometric identity for the product of sines: $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$. Applying this identity, we get $\frac{1}{2}[\cos(0) - \cos(18x)] = \frac{1}{2}[1 - \cos(18x)]$, which simplifies to $\sin^2 9x - \sin^2 9y$ using the identity $\sin^2 x = \frac{1}{2}(1 - \cos 2x)$.

Simplify the Expression Algebraically sin⁡(9x+9y)sin⁡(9x−9y)\sin ( 9 x + 9 y ) \sin ( 9 x - 9 y )sin(9x+9y)sin(9x−9y)

Simplify the Expression Algebraically $\sin ( 9 x + 9 y ) \sin ( 9 x - 9 y )$