Question

Prove that $4({\cos }^3{20}^o+{\cos }^3{40}^o)=3(\cos {20}^o+\cos {40}^o)$

Answer 1

$co{s}^{3}20+co{s}^{3}40$

$=(cos20+cos40)(co{s}^{2}20+co{s}^{2}40-cos20cos40)$

$=(cos20+cos40)\left(co{s}^2\right(30-10)+co{s}^2(30+10)-cos(30-10\left)cos\right(30+10\left)\right)$

$=(cos20+cos40)(cos30cos10+sin30sin10{)}^2-(cos30cos10+sin30sin10)$

$(cos30cos10-sin30sin10)+(cos30cos10-sin30sin10{)}^2$

$=(cos20+cos40)(co{s}^{2}30co{s}^{2}10+2sin10sin30cos10cos30+si{n}^{2}30si{n}^{2}10-co{s}^{2}30co{s}^{2}10-si{n}^{2}30si{n}^{2}10+co{s}^{2}30co{s}^{2}10-2sin10sin30cos10cos30+si{n}^{2}30si{n}^{2}10)$

$=(cos20+cos40)\times \frac{3}{4}\times (co{s}^{2}10+si{n}^{2}10)$

$=\frac{3}{4}(cos20+cos40)\times 1$

$=\frac{3}{4}(cos20+cos40)$

$\therefore 4\times (co{s}^{3}20\ast co{s}^{2}40)$

$=4\times \frac{3}{4}(cos20+cos40)$

$\Rightarrow 3(cos20+cos40)$

Hence, proved