Question

to prove cos3a+sin3a/cosa-sina=1+2sin2a

Answer 1

We know that cos3theta = 4costheta - 3cos^3 theta 
 
                        sin3theta = 3sintheta - 4sin^3theta


= 4cos^a - 3cosa + 3sina - 4sin^3a/cosa-sina 

= 4(cos^3a - sin^3a) - 3(cosa-sina)/cosa-sina

= 4(cos^3a - sin^3a)/cos a - sin a -  3

= 4(cos a - sin a)(cos^2 a +cos a sin a + sin ^2 a)/cos a - sin a - 3

= 4(1+cos a sin a) - 3

= 1 + 2 * 2 cos a sin a

= 1 + 2 sin 2a. 


Hope this helps!