sin2x/1+cos2x=1-cos2x/sin2x

Question

How do you prove it? Can you show the process of proving?

Answer 1

Please liberally use parentheses to state the problem exactly.

 

For example: ((sin(2x))/1) + (cos(2x)) =? 1 - ((cos(2x))/(sin(2x))) is what you have written.

 

I think what you mean is: (sin(2x))/(1+(cos(2x))) =? (1 - (cos(2x)))/(sin(2x)).

I'll prove this one.

 

Multiplying top and bottom of left side by the conjugate of (1+(cos(2x))):

((sin(2x))*(1 - (cos(2x))))/((1+(cos(2x)))*(1 - (cos(2x))))

Multiplying the two factors in the denominator:

((sin(2x))*(1 - (cos(2x))))/(1 - (cos(2x))^2)

Substituting the known identity (1 - (cos(2x))^2) = ((sin(2x))^2):

((sin(2x))*(1 - (cos(2x))))/((sin(2x))^2)

Dividing numerator and denominator by (sin(2x)):

(1 - (cos(2x))))/(sin(2x)) = right side. Q.E.D.

 

Notice that the 2x could have been just x (or theta, etc). The 2 is just thrown in to confuse you; you never had to deal with it by using double angle identities.