How to Derive (cos x)^2 and (sin x)^2 Using Trig Identities

[frasifrasi]

I was wondering how my book got from

8.cos^(2)(x)

to 4(1+ cos(2x))

What trig ID is this?

thank you.

 

[dynamicsolo]

The identity for (cos x)^2 is constructed from the double-angle formula for cosine,

cos(2x) = (cos x)^2 - (sin x)^2,

together with the "Pythagorean Identity",

(cos x)^2 + (sin x)^2 = 1 .

 

[frasifrasi]

But, how did it become 4(1+ cos(2x))?

 

[dynamicsolo]

cos(2x) = (cos x)^2 - (sin x)^2 ;

cos(2x) = (cos x)^2 - [ 1 - (cos x)^2 ] = 2·[ (cos x)^2 ] - 1 ;

(cos x)^2 = (1/2) · [ 1 + cos(2x) ]

This is a good derivation to keep in mind for finding (cos x)^2 and (sin x)^2 ; it is the "trick" used when we need to integrate those functions...