(cos5x+cos7x)^2=(sin5x+sin7x)^2

(cos^25x + 2cos5xcos7x + cos^27x) = (sin^25x + 2sin5xsin7x + sin^27x)

Expanding both sides:

cos^25x + 2cos5xcos7x + cos^27x = sin^25x + 2sin5xsin7x + sin^27x

Now, we know that cos^2x + sin^2x = 1

So, we can replace cos^2 with 1 - sin^2 on the left side:

(1 - sin^25x) + 2cos5xcos7x + (1 - sin^27x) = sin^25x + 2sin5xsin7x + sin^27x

Now, we can simplify:

2cos5xcos7x = 2sin5xsin7x

Dividing both sides by 2cos5xcos7x:

1 = tan5xtan7x

Therefore, the solution to the equation (cos5x+cos7x)^2=(sin5x+sin7x)^2 is x = n pi + arctan(1/tan(5npi)tan(7n*pi)), where n is an integer.