- #1
Matt Jacques
- 81
- 0
What simpler indentity is equal to sin(x) - cos(x) ?
Trig Identities have come back to haunt me!
Trig Identities have come back to haunt me!
Originally posted by Matt Jacques
What simpler indentity is equal to sin(x) - cos(x) ?
Trig Identities have come back to haunt me!
Originally posted by mathman
cos(x+y)=cos(x)cos(y)-sin(x)sin(y). Lety=45o. Net result
sin(x)-cos(x)=-sqrt(2)cos(x+y).
Is that simple enough?
The simpler identity for sin(x) - cos(x) is tan(x/2), also known as the half-angle formula for tangent.
The half-angle formula for tangent, tan(x/2), is derived from trigonometric identities involving sin(x) and cos(x), making it equivalent to sin(x) - cos(x).
No, tan(x/2) cannot be simplified further as it is already the simplest identity equivalent to sin(x) - cos(x).
Tan(x/2) is preferable over sin(x) - cos(x) when dealing with half-angle problems, when finding the value of an angle that is half of a given angle. It is also commonly used in integration and differentiation problems.
The half-angle formula for tangent, tan(x/2) = (sin(x))/(1+cos(x)), can be verified by using the double-angle formula for cosine, cos(2x) = 1 - 2sin^2(x), and substituting x/2 for x. This will result in tan(x/2) = (2sin(x/2)cos(x/2))/(1+cos(x)), which can be simplified to tan(x/2) = sin(x) - cos(x).