Question

How does one prove the following equation? (Image attached for reference)

Answer 1

See below

Explanation:

Since I'm given 4 blanks, here's my approach:

`tan(theta/2)= sintheta/(1+costheta)`

1. `sin(theta/2)/cos(theta/2)= sintheta/(1+costheta)`
   Reason: Quotient Identity : `sintheta/costheta= tantheta`

2. `(sqrt((1-costheta)/2))/(sqrt((1+costheta)/2))= sintheta/(1+costheta)`
   Reason: Half angle Identities : `sin(theta/2)=sqrt((1-costheta)/2)` and `cos(theta/2)=sqrt((1+costheta)/2)`

3. `(sqrt((1-costheta)/cancel2))*(sqrt(cancel2/(1+costheta)))= sqrt((1-costheta)/(1+costheta))=sintheta/(1+costheta)`

Reason: Division of fractions

1. `sqrt((1-costheta)/(1+costheta)*(1+costheta)/(1+costheta))= sqrt((1-cos^2theta)/(1+costheta)^2)=sqrt((sin^2theta)/(1+costheta)^2)= sintheta/(1+costheta)`

Reason: Rationalization of the denominator and modified Pythagorean identity: `1-cos^2theta= sin^2theta`