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

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1 Answer
Apr 8, 2018

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