How can I use the trigonometric subtraction formula with reasoning for sine to verify sin[(pi/2)-x]=cosx?

You need to use the following trigonometric subtraction formula to prove that `sin(pi/2 - x) = cos x` , such that:

`sin(a - b) = sin a*cos b - sin b*cos a`

Considering` a = pi/2` and ` b = x` , yields:

`sin(pi/2 - x) = sin(pi/2)*cos x - sin x*cos(pi/2)`

You need to remember that the value of sine function at `pi/2` is maximum, hence `sin(pi/2) = 1` . You also need to remember that `cos(pi/2) = 0` .

Replacing 1 for `sin(pi/2)` and 0 for `cos(pi/2)` in subtraction formula above, yields:

`sin(pi/2 - x) = 1*cos x - sin x*0`

Since the product  `sin x*0 = 0` , yields:

`sin(pi/2 - x) = cos x`

Hence, the equality `sin(pi/2 - x) = cos x ` is verified by using the subtraction trigonometric formula `sin(a - b) = sin a*cos b - sin b*cos a.`