Question

$1+\cos {56}^{\circ }+\cos {58}^{\circ }-\cos {66}^{\circ }=m\cos {28}^{\circ }\cos {29}^{\circ }\sin {33}^{\circ }$. Find m

Answer 1

$1+\cos 56+\cos 58-\cos 66=m\cos 20\cos 29\sin 33$

$=m\cos 28\cos 29\sin 33$

$=\frac{m}{2}\left[2\cos 28\cos 29\right]\sin 33$

$=\frac{m}{2}[\cos 57+\cos 1]\sin 33$

$=\frac{m}{4}[2\cos 57\sin 33+2\cos 1\sin 33]$

$=\frac{m}{4}\left[\sin \right(90)+(\sin 24)+\sin 34-\sin 32]$

$=\frac{m}{4}[1-\sin {24}^o+\sin {34}^o-\sin {32}^o]$

$=\frac{m}{4}[1-\cos {66}^o\cos {56}^o-\cos {58}^o]$

$\Rightarrow \frac{m}{4}=1\Rightarrow m=4$

Hence the value of $m$ is $4$