Use the trigonometric subtraction formula for sine to verify this identity: `sin(pi/2 - x) =cosx`   Please provide a reason with the 3 steps.

Expert Answers
lemjay eNotes educator| Certified Educator

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

To prove, at the left side, apply the identity for difference of two angles of sine which is `sin(A - B) = sinAcosB - sinBcosA` .


Since `sin (pi/2)=1` and `cos(pi/2)=0` , plug-in these values to the equation.

`1*cosx - sinx *0 = cosx`

`cosx - 0 = cos x`

`cosx=cos x`    (True)

Since simplifying left side results to a true condition, this proves that the given equation is an identity.

violy eNotes educator| Certified Educator

Use the difference between two angles for sine. 

`sin(a - b) = sinacosb - cosasinb`

So, we will have:

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

We know that sin(pi/2) = 1, and cos(pi/2) = 0.

Plug-in those values.

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


`1*cosx - 0*sinx = cosx`