Question

1)  sina+sin2a+sin3a+sin4a=?
2) cos2a-cos4a-cos6a+cos8a=?


3) cosa=1/ $sqrt{3}$
cos2a-cos6a=?
4) sina=2/ $sqrt{5}$
sin5a-sin3a=?

Answer 1

$(sina+sin4a)+(sin2a+sin3a) = 2sinfrac{5a}{2}*cosfrac{3a}{2}+2sinfrac{5a}{2}*cosfrac{a}{2}\ 2sinfrac{5a}{2}(cosfrac{3a}{2}+cosfrac{a}{2})$

$(cos2a+cos8a)-(cos4a+cos6a)=\ 2cos5a*cos3a-(2cos5a*cosa)=2cos5a(cos3a-cosa)$

$cosa=frac{1}{sqrt{3}}=frac{sqrt{3}}{3}\ sina=frac{sqrt{6}}{{3}}\ cos2a-cos6a=16sina*cosa*(2cos^2a-1)*sina*cosa=\ 16*frac{sqrt{18}}{3}*(2*frac{1}{3}-1)*frac{sqrt{18}}{3}= -frac{32}{3}$

$sina=frac{2}{sqrt{5}}\ sin5a-sin3a=\ 2sina*(2(1-2sin^2a)^2-1)=frac{4}{sqrt{5}}(2* frac{9}{25} -1) = frac{4}{sqrt{5}}*-frac{7}{25}=\ -frac{28}{25sqrt{5}}$