To check option #5, we substitute x = 37.6º in the equation:
sin(x) = cos(x - π/2)
sin(37.6º) = cos(37.6º - π/2)
Using the angle subtraction identity cos(a - b) = cos(a)cos(b) + sin(a)sin(b), we have:
sin(37.6º) = cos(37.6º)cos(π/2) + sin(37.6º)sin(π/2)
sin(37.6º) = cos(37.6º)(0) + sin(37.6º)(1)
sin(37.6º) = sin(37.6º)
Since both sides of the equation are equal, option #5 is true.
Therefore, the two options that are true for all values of x are:
1) cos(x) = cos(x - π/2)
5) sin(x) = cos(x - π/2)