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Solve the Trigonometry Identity

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2 Answers

cosx = (1- tan^2 x/2) / 1 + tan^2 x/2)
cosx= (1- tan^2 x/2) / sec^2 x/2=(1- tan^2 x/2)cos^2 x/2=cos^2 x/2-sin^2 x/2=cos x
 
tan(x/2)=sin(x/2)/cos(x/2), by definition.
 
1-tan2(x/2)=1-sin2(x/2)/cos2(x/2)=[cos2(x/2)-sin2(x/2)]/cos2(x/2);
 
Now remember that cos(a+b)=cos(a)cos(b)-sin(a)sin(b) If a=b=x then cos(2x)=cos2(x)-sin2(x). Apply this to the expression above to get:
1-tan2(x/2)=cos(x)/cos2(x/2);
 
Now similarly we can get:
 
1+tan2(x/2)=1+sin2(x/2)/cos2(x/2)=[cos2(x/2)+sin2(x/2)]/cos2(x/2);
 
Now recall basic trigonometric identity: cos2(x)+sin2(x)=1; Thus we obtain:
 
1+tan2(x/2)=1/cos2(x/2);
 
Finally, [1-tan2(x/2)]/[1+tan2(x/2)]=[cos(x)/cos2(x/2)]/[1/cos2(x/2)]=cos(x), since cos2(x/2) factors cancel out.