Prove Trigonometric Identities

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David Cohen questions just for fun.
Enjoy.

IMG_20220205_215330.jpg
 
MSP374523ffe0dhb2a31a94000043f1g5406g1a7255
is not true statement


tan(x/2)=(1-cos(x)+sin(x))/(1+cos(x)+sin(x))

manipulate right side
(1-cos(x)+sin(x))/(1+cos(x)+sin(x))............ 1-cos(x)=2sin^2(x/2), 1+cos(x)=2cos^2(x/2)

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

=(2 sqrt(2) sin(x/2 + π/4) sin(x/2))/(2sqrt(2) sin(x/2 + π/4) cos(x/2))

= sin(x/2)/cos(x/2)

= tan(x/2)
 
MSP374523ffe0dhb2a31a94000043f1g5406g1a7255
is not true statement


tan(x/2)=(1-cos(x)+sin(x))/(1+cos(x)+sin(x))

manipulate right side
(1-cos(x)+sin(x))/(1+cos(x)+sin(x))............ 1-cos(x)=2sin^2(x/2), 1+cos(x)=2cos^2(x/2)

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

=(2 sqrt(2) sin(x/2 + π/4) sin(x/2))/(2sqrt(2) sin(x/2 + π/4) cos(x/2))

= sin(x/2)/cos(x/2)

= tan(x/2)

Don't you just love that Cohen Textbook?
Can you set up the second trigonometric identity for me to try on my own?

How about this one?

Find Speed
 
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