Write the difference in tanpi/4-cosx as a product?
We have to write tan ( pi/4) - cos x as a product
We know that tan x = sin x / cos x and tan (pi/4) = 1
tan ( pi/4) - cos x
=> 1 - cos x
we can write 1 as cos 0
=> cos 0 - cos x
=> −2 sin ( 0 + x)/2 * sin (0 - x)/2
=> -2* sin x/2 * sin -x/2
=> 2* sin x/2 * sin x/2
=> 2* (sin x/2)^2
Therefore the required result is 2* (sin x/2)^2
First, we'll substitute the function tan pi/4 by it's value 1.
To transform the difference into a product, we'll have to express the value 1 as being the function cosine of an angle, so that the terms of the difference to be 2 like trigonometric functions.
1 = cos 0 or cos 2pi
1 - cosx = cos 0-cos x
cos 0-cos x = -2 sin (0+x)/2*sin (0-x)/2
cos 0-cos x = -2sin (x/2)*sin (-x/2)
Because of the fact that the trigonometric function sine is an odd function , we'll write sin (-x/2)=-sin (x/2)
cos 0-cos x = 2sin (x/2)*sin (x/2)
tan pi/4 - cos x = 2[sin (x/2)]^2