Solve the Trigonometry Identity

Question

cosx = (1- tan^2 x/2)  /  1 + tan^2 x/2)

Completely stuck.

Answer 1

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

 

Answer 2

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.