Given:
#cos3theta= 4cos^3theta-3costheta#
#sin3theta= 3sintheta-4sin^3theta#
Apply the quotient identity:
#(cos3theta)/(sin3theta)= cot3theta#
#(cos3theta)/(sin3theta)= (t^3-3t)/(3t^2-1)#
Substitute givens from above:
#(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)=(t^3-3t)/(3t^2-1)#
Substitute #cottheta# for #t#:
#(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)=((cottheta)^3-3(cottheta))/(3(cottheta)^2-1)#
Simplify:
#(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)=(cot^3theta-3cottheta)/(3cot^2theta-1)#
Apply quotient identities:
#(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)=(cos^3theta/sin^3theta-(3costheta)/sintheta)/((3cos^2theta)/sin^2theta-1)#
Common denominator:
#(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)=(cos^3theta/sin^3theta-(3costhetasin^2theta)/sin^3theta)/((3cos^2theta)/sin^2theta-sin^2theta/sin^2theta)#
Combine and divide:
#(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)=((cos^3theta-3costhetasin^2theta)/sin^3theta)/((3cos^2theta-sin^2theta)/sin^2theta#
#(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)=(costheta(cos^2theta-3sin^2theta))/sin^3theta*sin^2theta/(3cos^2theta-sin^2theta#
#(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)=(costheta(cos^2theta-3sin^2theta))/sintheta*1/(3cos^2theta-sin^2theta#
Apply Pythagorean identities #sin^2theta= 1-cos^2theta and cos^2theta= 1-sin^2theta# :
#(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)=(costheta(cos^2theta-3(1-cos^2theta)))/(sintheta*(3(1-sin^2theta)-sin^2theta)#
Simplify:
#(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)=(costheta(cos^2theta-3+3cos^2theta))/(sintheta*(3-3sin^2theta-sin^2theta)#
#(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)=(cos^3theta-3costheta+3cos^3theta)/(3sintheta-3sin^3theta-sin^3theta)#
Combine like terms:
#(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)=(4cos^3theta-3costheta)/ (3sintheta-4sin^3theta)#
VERIFIED!
P.S. Almost thought I was going nowhere
