Understanding the Trigonometric Identity: cos^2 x = 1/2 + cos(2x)/2

[leopard]

Why is $$cos^2 x = \frac{1}{2} + \frac{cos(2x)}{2}$$

?

 

[lurflurf]

why is
[cos(x)]^2=1-[sin(x)]^2
why is
[cos(x)]^2=cos(2x)+[sin(x)]^2

 

[HallsofIvy]

Starting where? Do you know cos(a+ b)= cos(a)cos(b)- sin(a)sin(b)? If so then take a= b= x. If not then what definition of cos(x) are you using so that you can get that?

 

[Дьявол]

I think this is what you wanted to know:
cos²x+sin²x=1
cos²x=1-sin²x
Since cos(2x)=cos²x-sin²x
cos²x=1+cos(2x)-cos²x
2cos²x=1+cos(2x)
cos²x=1/2 + cos(2x)/2

 

[icystrike]

Why is $$cos^2 x = \frac{1}{2} + \frac{cos(2x)}{2}$$

?

RHS:$$\frac{1}{2} + \frac{cos(2x)}{2}$$

$$=\frac{1+cos(2x)}{2}$$

$$=\frac{1+2cos^{2}x-1}{2}$$

$$=\frac{2cos^{2}x}{2}$$

$$=cos^{2}x$$

$$=LHS(Shown)$$