Question

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

Answer 1

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`