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To simplify the expression ( cos(36^circ + A) cos(36^circ – A) + cos(54^circ + A) cos(54^circ – A) ), we can use the cosine addition formula:
[
cos(x + y) cos(x – y) = frac{1}{2} left( cos(2x) + cos(2y) right)
]
Applying this formula to each term:
1. For ( cos(36^circ + A) cos(36^circ – A) ):
[
cos(36^circ + A) cos(36^circ – A) = frac{1}{2} left( cos(72^circ) + cos(2A) right)
]
2. For ( cos(54^circ + A) cos(54^circ – A) ):
[
cos(54^circ + A) cos(54^circ – A) = frac{1}{2} left( cos(108^circ) + cos(2A) right)
]
Now, we can combine these results:
[
cos(36^circ + A) cos(36^circ – A) + cos(54^circ + A) cos(54^circ