Maisarah S.
asked • 02/27/21cos2 (x/2) = (tanx + sinx) / 2 tanx
Dayv O. answered • 02/27/21
Caring Super Enthusiastic Knowledgeable Trigonometry Tutor
Maisarah,
you are right, this is a proven Identity valid for any x..
I know cos2x=(1/2)*(1+cos2x) [from cos2x=cos2x-sin2x, and sin2x=1-cos2x
so cos2(x/2)=(1/2)*(1+cosx)
cos2 (x/2) = (tanx + sinx) / 2 tanx
tanx/2tanx=1/2
sinx/2tanx=(cosx)/2
Sam Z. answered • 02/27/21
Math/Science Tutor
x=45°.......................?
Sam Z.
Give me another answer.02/27/21
Dayv O.
x=1000 radians (=about 57000 degrees) cos^2(1000/2)=.78 tan 10000 = 1.47 sin 1000 =.83 (1.47+.83)/(2*1.47)=.7802/27/21
Sam Z.
ok; I used 1 rad and got-1.09786......02/27/21
Dayv O.
cos^2(1/2)=.77 tan1=1.56 sin1=.84 (1.56+.84)/(2*1.56)=.77 for all values of x, cos^2(x/2)=(tanx+sinx)/2tanx02/28/21
Dayv O.
if you notice, kind of x=0 is a question as 0/0 might or might not be equal to 1(equal cos^2(0). Like a lot of 0/0 (all derivatives in calculus) the discontinuity implied by dividing by zero in this case is a removable discontinuity. By using the calculus L'Hopital rule the limit when x approaches zero of (tanx+sinx)/2*tanx is 1 just like cos^2(0) equals. so the identity works for all x and 0/0 is 1 in this instance. cos(x/2) is a continuous function even at x=0. yay as they say.02/28/21
Sam Z.
JUST HAVE TO MENTION: When x= 45; the outcome is -2*10^-12.02/28/21
Dayv O.
Sam,,, cosine squared of (45 degrees/2) is .854.;;;;;;;;;;;;;;;;;;;;;If you check what (tan(45)+sin(45))/2*tan(45) you will find it equals .85402/28/21
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Dayv O.
please see my answer, yes x=45 degrees works as well as any x.02/27/21