Write the difference in tanpi/4-cosx as a product?  

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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

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