Question

cos(alpha-3theta)/cos^3 theta = sin(alpha-3theta)/sin^3 theta = m
prove that, cos alpha = (2-m^2)/m​

Answer 1

Answer:

cos alpha value is coming out to be (2 – m^2)/m.

obviously you can find cos2alpha by using cos2alpha= 2cos^2alpha – 1 in terms of m.

to prove that cos alpha= (2 – m^2)/m, you need to note that Cos (alpha-3theta)= mcos3 theta and sin (alpha- 3theta)= msin3theta so square and add to get

1= m^2(sin^6theta + cos^6theta)= m^2(1 – 3sin^2(2theta)/4)= m^2(3cos4theta+5)/8.......(1)

now, from Cos (alpha-3theta)= mcos3 theta and sin (alpha- 3theta)= msin3theta, we can expand to write

cosalpha*cos3theta + sinalpha*sin3theta= mcos3 theta

and sinalpha*cos3theta – sin3theta*cosalpha= msin3theta

from these two eqns, eliminate sinalpha to obtain the value of cosalpha in terms of theta.

after simplifying, you obtain

cosalpha= m(1+3cos4theta)/4.......(2)

from (1) and (2), eliminate cos4theta to obtain the answer.

kindly approve :)