How do you use the composite argument property to show that an equation is an identity?

Question

Use the composite properties to show that sin(θ+90) - cos(θ+180)= 2cosθ is an identity.

Answer 1

First you must employ the composite argument properties that are applicable to the terms in the equation. Here, the left-hand side (LHS) consists of two such terms: sin(θ+90) and cos(θ+180). We will transform LHS into the right-hand side (RHS) of the equation to show equality.

The first is of the form sin(a+b) which has the composite argument a+b and is equal to sin(a)*cos(b)+cos(a)*sin(b).

The second is of the form cos(a+b) which also has the composite argument a+b and is equal to cos(a)*cos(b)-sin(a)*sin(b).

Thus:

LHS = [ sin(θ)*cos(90)+cos(θ)*sin(90) ] - [ cos(θ)*cos(180)-sin(θ)*sin(180) ]

= sin(θ)*cos(90) + cos(θ)*sin(90) - cos(θ)*cos(180) + sin(θ)*sin(180)

(factoring out sin(θ) and cos(θ))

= sin(θ) * [cos(90) + sin(180)] + cos(θ) * [sin(90) - cos(180)]

(with: cos(90)=0=sin(180), sin(90)=1, cos(180)=-1)

= sin(θ) * [ 0 + 0 ] + cos(θ) * [ 1 - (-1) ]

= cos(θ) * 2 = 2cos(θ) = RHS.

QED.