Question

If $a,b,c$ and $d$ are four numbers in the interval $\left[0,\pi \right]$ such that $sina+7sinb=4\left(sinc+2sind\right)$ and $cosa+7cosb=4\left(cosc+2cosd\right)$ , then the numerical value of $\frac{7cos\left(b-c\right)}{cos\left(a-d\right)}$ is

Answer 1

Answer: 2

It is given that
$sina+7sinb=4sinc+8sind$
$\Rightarrow sina-8sind=4sinc-7sinb....\left(i\right)$
Also given that
$cosa-8cosd=4cosc-7cosb\dots \left(ii\right)$
Squaring and adding Eq. (i) and (ii), we get
$1+8^2-16\left(sinasind+cosacosd\right)$
$=16+49-56\left(sinbsinc+cosbcosc\right)$
$\Rightarrow -16cos\left(a-d\right)=-56cos\left(b-c\right)$
$\Rightarrow \frac{cos\left(b-c\right)}{cos\left(a-d\right)}=\frac{2}{7}$
$\Rightarrow \frac{7cos\left(b-c\right)}{cos\left(a-d\right)}=2$