To solve the equation sin(3x)sin(x) - cos(7x)cos(5x) = 0, we can use the product-to-sum identities for sine and cosine, which state that:
sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)]cos(A)cos(B) = 1/2[cos(A-B) + cos(A+B)]
Applying these identities to the given equation, we have:
1/2[cos(3x-x) - cos(3x+x)] - 1/2[cos(7x+5x) + cos(7x-5x)] = 01/2[cos(2x) - cos(4x)] - 1/2[cos(12x) + cos(2x)] = 01/2cos(2x) - 1/2cos(4x) - 1/2cos(12x) - 1/2cos(2x) = 0-1/2cos(4x) - 1/2cos(12x) = 0
Now we can simplify this to:
-1/2cos(4x) - 1/2cos(12x) = 0cos(4x) + cos(12x) = 0
However, this equation does not immediately yield a solution since the sum of two cosine functions may not necessarily be zero. You may need to further simplify or revise your approach to finding solutions.
To solve the equation sin(3x)sin(x) - cos(7x)cos(5x) = 0, we can use the product-to-sum identities for sine and cosine, which state that:
sin(A)sin(B) = 1/2[cos(A-B) - cos(A+B)]
cos(A)cos(B) = 1/2[cos(A-B) + cos(A+B)]
Applying these identities to the given equation, we have:
1/2[cos(3x-x) - cos(3x+x)] - 1/2[cos(7x+5x) + cos(7x-5x)] = 0
1/2[cos(2x) - cos(4x)] - 1/2[cos(12x) + cos(2x)] = 0
1/2cos(2x) - 1/2cos(4x) - 1/2cos(12x) - 1/2cos(2x) = 0
-1/2cos(4x) - 1/2cos(12x) = 0
Now we can simplify this to:
-1/2cos(4x) - 1/2cos(12x) = 0
cos(4x) + cos(12x) = 0
However, this equation does not immediately yield a solution since the sum of two cosine functions may not necessarily be zero. You may need to further simplify or revise your approach to finding solutions.