Simplify the expression algebraically. $$\cos ( 7 x + 4 y ) \cos ( 7 x - 4 y )$$
A) $\cos ^ { 2 } 7 x - \sin ^ { 2 } y$
B) $\cos ^ { 2 } 7 x + \sin ^ { 2 } 4 y$
C) $\cos ^ { 2 } 7 x - \sin ^ { 2 } 4 y$
D) $\cos ^ { 2 } x - \sin ^ { 2 } 4 y$
E) $\cos ^ { 2 } x - \sin ^ { 2 } y$

Question

Simplify the expression algebraically.​ $$\cos ( 7 x + 4 y ) \cos ( 7 x - 4 y )$$ ​
A) $\cos ^ { 2 } 7 x - \sin ^ { 2 } y$
B) $\cos ^ { 2 } 7 x + \sin ^ { 2 } 4 y$
C) $\cos ^ { 2 } 7 x - \sin ^ { 2 } 4 y$
D) $\cos ^ { 2 } x - \sin ^ { 2 } 4 y$
E) $\cos ^ { 2 } x - \sin ^ { 2 } y$

Quizplus · Accepted Answer

This expression can be simplified using the trigonometric identity for the product of cosines: $\cos A \cos B = \frac{1}{2}[\cos(A + B) + \cos(A - B)]$. Applying this to the given expression, we get $\frac{1}{2}[\cos((7x + 4y) + (7x - 4y)) + \cos((7x + 4y) - (7x - 4y))]$, which simplifies to $\frac{1}{2}[\cos(14x) + \cos(8y)]$. However, this approach was incorrect in identifying the specific form matching the provided options. The correct simplification, aligning with the options, actually leverages a different identity or approach that directly leads to the form seen in option C, which correctly represents the product of cosines in terms of squared functions. The correct identity to use is $\cos^2 A - \sin^2 B$, where $A = 7x$ and $B = 4y$, leading directly to the correct answer without the need for the intermediate step I incorrectly described.

Simplify the Expression Algebraically cos⁡(7x+4y)cos⁡(7x−4y)\cos ( 7 x + 4 y ) \cos ( 7 x - 4 y )cos(7x+4y)cos(7x−4y)

Simplify the Expression Algebraically $\cos ( 7 x + 4 y ) \cos ( 7 x - 4 y )$