How do you write #2cos^2 5-1# as a single trigonometric function?

1 Answer
Jul 24, 2016

cos10

Explanation:

Using the basic #color(blue)"double angle expansion for cosine"#
We can develop further expansions.

#color(red)(|bar(ul(color(white)(a/a)color(black)(cos2x=cos^2x-sin^2x)color(white)(a/a)|)))........ (A)#

along with #color(red)(|bar(ul(color(white)(a/a)color(black)(sin^2x+cos^2x=1)color(white)(a/a)|)))........ (B)#

From (B) we can obtain.

#sin^2x=1-cos^2x" and " cos^2x=1-sin^2x#

Substitute these in turn into right side of (A)

#rArr1-sin^2x-sin^2x=1-2sin^2x#

and #cos^2x-(1-cos^2x)=2cos^2x-1#

#rArrcos2x=cos^2x-sin^2x=1-2sin^2x=2cos^2x-1#

Using the identity #cos2x=2cos^2x-1#

#rArr2cos^2 5-1=cos(2xx5)=cos10#