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Verify the identity= (cos x/2-sin x/2)^2=1-sin x

 

can someone show work using the reciprocal identies.....struggling :(

 Dec 16, 2019
 #1
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 (cos x/2-sin x/2)^2 = 1-sin x

 

Work with the left side

 

[ cos (x/2)  - sin (x/2)] ^2  =

 

{cos (x/2) - sin(x/2) ] * [ cos(x/2) - sin (x/2) ]  =

 

cos^2 (x/2)  - 2sin(x/2)cos(x/2)  + sin^2 (x/2)

 

Let x/2   = θ     ......and we have that

 

cos^2 (θ)  -  2sin(θ)cos(θ)  + sin^2(θ)   =

 

sin^2(θ)  + cos^2(θ)  - 2sin(θ)cos(θ)  =

 

1   -  sin (2θ)    .....now.....back-substitute

 

1  - sin [ 2  (x/2) ]   =

 

1 - sin x         which =  the right side  !!!

 

cool cool cool

 Dec 16, 2019

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