Question

${lim}_{x\to 0}\frac{cosax-cosbx}{coscx-cosdx}$

Answer 1

${lim}_{x\to 0}\frac{cosax-cosbx}{coscx-cosdx}$

Applying Componendo & Dividendo formula we get

$={lim}_{x\to 0}\frac{-2sin\left(\frac{a+b}{2}\right)xsin\left(\frac{a-b}{2}\right)x}{-2sin\left(\frac{c+d}{2}\right)xsin\left(\frac{c-d}{2}\right)x}=\frac{{lim}_{x\to 0}sin\left(\frac{a+b}{2}\right)\times {lim}_{x\to 0}sin\left(\frac{a+b}{2}\right)x}{{lim}_{x\to 0}sin\left(\frac{c+d}{2}\right)\times {lim}_{x\to 0}sin\left(\frac{c-d}{2}\right)x}$

$=\frac{({lim}_{x\to 0}\frac{sin\left(\frac{a+b}{2}\right)x}{\left(\frac{a+b}{2}\right)x}\times \left(\frac{a+b}{2}\right)x)({lim}_{x\to 0}\frac{sin\left(\frac{a-b}{2}\right)x}{\left(\frac{a-b}{2}\right)x}\times \left(\frac{a-b}{2}\right)x)}{({lim}_{x\to 0}\frac{sin\left(\frac{c+d}{2}\right)x}{\left(\frac{c+d}{2}\right)x}\times \left(\frac{c+d}{2}\right)x)({lim}_{x\to 0}\frac{sin\left(\frac{c-d}{2}\right)x}{\left(\frac{c-d}{2}\right)x}\times \left(\frac{c-d}{2}\right)x)}$

$=\frac{(a+b)(a-b)}{(c+d)(c-d)}[\because {lim}_{0\to 0}\frac{sin\theta }{\theta }=1]$

$=\frac{a^2-b^2}{c^2-d^2}$