Question

In triangle ABC, right-angled at B, if $\tan A=\frac{1}{\sqrt{3}}$, find the value of:
(i) $\sin A\cos C+\cos A\sin C$
(ii) $\cos A\cos C-\sin A\sin C$

Answer 1

$\angle B={90}^0=\pi /2$

So, $\left(I\right)$ $sinAcosC+cosAsinC=\sin (A+C)=\sin (\pi -B)$

$=sinB=\sin \pi /2=1$

$\left(II\right)$ $cosAcosC-sinAsinC=\cos (A+C)=\cos (\pi -B)$

$=-cosB=-\cos \left(\pi /2\right)=0$