Question

plz solve this...........​

Answer 1

$\underline{\text{Correcting the question:}}$

$\mathrm{Now,\:\frac{cos3\alpha+2\:cos5\alpha+cos7\alpha}{cos\alpha+2\:cos3\alpha+cos5\alpha}}$

$\mathrm{=\frac{(cos7\alpha+cos3\alpha)+2\:cos5\alpha}{(cos5\alpha+cos\alpha)+2\:cos3\alpha}}$

$\mathrm{=\frac{2\:cos\frac{7\alpha+3\alpha}{2}\:cos\frac{7\alpha-3\alpha}{2}+2\:cos5\alpha}{2\:cos\frac{5\alpha+\alpha}{2}\:cos\frac{5\alpha-\alpha}{2}+2\:cos3\alpha}}$

$\mathrm{=\frac{2\:cos5\alpha\:cos2\alpha+2\:cos5\alpha}{2\:cos3\alpha\:cos2\alpha+2\:cos3\alpha}}$

$\mathrm{=\frac{2\:cos5\alpha(cos2\alpha+1)}{2\:cos3\alpha(cos2\alpha+1)}}$

$\mathrm{=\frac{cos5\alpha}{cos3\alpha}}$

$\to \boxed{\small{\mathrm{\frac{cos3\alpha+2\:cos5\alpha+cos7\alpha}{cos\alpha+2\:cos3\alpha+cos5\alpha}=\frac{cos5\alpha}{cos3\alpha}}}}$

$\text{Hence, proved.}$

$\underline{\text{Rules :}}$

$\mathrm{1.\:sinC+sinD=2\:sin\frac{C+D}{2}\:cos\frac{C-D}{2}}$

$\mathrm{2.\:cosC+cosD=2\:cos\frac{C+D}{2}\:cos\frac{C-D}{2}}$

$\underline{\text{As per the question :}}$

$\mathrm{Now,\:\frac{cos3\alpha+2\:cos5\alpha+cos7\alpha}{sin3\alpha+2\:sin5\alpha+sin7\alpha}}$

$\mathrm{=\frac{(cos7\alpha+cos3\alpha)+2\:cos5\alpha}{(sin7\alpha+sin3\alpha)+2\:sin5\alpha}}$

$\mathrm{=\frac{2\:cos\frac{7\alpha+3\alpha}{2}\:cos\frac{7\alpha-3\alpha}{2}+2\:cos5\alpha}{2\:sin\frac{7\alpha+3\alpha}{2}\:cos\frac{7\alpha-3\alpha}{2}+2\:sin5\alpha}}$

$\mathrm{=\frac{2\:cos5\alpha\:cos2\alpha+2\:cos5\alpha}{2\:sin5\alpha\:cos2\alpha+2\:sin5\alpha}}$

$\mathrm{=\frac{2\:cos5\alpha(cos2\alpha+1)}{2\:sin5\alpha(cos2\alpha+1)}}$

$\mathrm{=\frac{cos5\alpha}{sin5\alpha}}$

$\mathrm{=cot5\alpha}$

$\to \boxed{\small{\mathrm{\frac{cos3\alpha+2\:cos5\alpha+cos7\alpha}{sin3\alpha+2\:sin5\alpha+sin7\alpha}=cot5\alpha}}}$

$\text{This is the required simplification.}$