# Cosine identity

#### Giestforlife

I came across the following while doing integration:

I don't need a proof, I just want to understand what general property (law, theorem, identity?) of trig that is at work here.

Thanks,

Giest

#### zhandele

Think of the unit circle. You'll see that cos(x + pi) = - cos(x). So cos(x) - cos(x + pi) = cos(x) - (-cos(x)) = 2 cos(x). Then divide that by 2.

1 person

#### Shakarri

If you draw the unit circle and label the quadrants with which trig values are positive it is easy to visualise.

You can see that adding 180 degrees onto an angle brings a point on the circle from a positive quadrant to a negative quadrant or vice versa. But when you add 180 degrees to a point it remains the same angle from the horizontal axis.
Taking these two factors into account you can see that cos(x)= -cos(x+180)

1 person

#### HallsofIvy

MHF Helper
cos(x+ y)= cos(x)cos(y)- sin(x)sin(y)

So $$\displaystyle cos(x+\pi)= cos(x)cos(\pi)- sin(x)sin(\pi)= cos(x)(-1)+ sin(x)(0)= -cos(x)$$

1 person

#### Giestforlife

Thanks to those who answered! An absurdly simple explanation.