ADDITION OF SINE AND COSINE
(1) We can use trig identities to bring the two components together. First let’s generalise then work towards specifics.
sin(a+b)=sin(a)cos(b)+cos(a)sin(b),
sin(a-b)=sin(a)cos(b)-cos(a)sin(b).
If we add these identities we get:
sin(a+b)+sin(a-b)=2sin(a)cos(b).
We can also use a relationship between sine and cosine:
sin(a-b)=cos(90-(a-b))=cos(90-a+b).
Therefore, sin(a+b)+cos(90-a+b)=2sin(a)cos(b).
Let x=a+b and y=90-a+b.
So x+y=90+2b, making b=½(x+y)-45.
From this a=x-b=½(x-y)+45.
So sin(x)+cos(y)=2sin(½(x-y)+45)cos(½(x+y)-45).
That’s the general case. Now to find sin(x)+cos(x) by plugging in y=x:
sin(x)+cos(x)=2sin(45)cos(x-45)=√2cos(x-45).
Now we use the fact that cos(θ)=cos(-θ) (cosine is an even function).
So sin(x)+cos(x)=√2cos(45-x).
cos(45-x)=sin(90-(45-x))=sin(x+45), and:
sin(x)+cos(x)=√2sin(x+45).