Question

Evaluate $\sec {70}^o\sin {20}^o-\cos {70}^o\csc {20}^o$

Answer 1

Consider $\sin {20}^0$ and $\sec {70}^0$ which can be rewritten as follows:

$\sin {20}^0=\sin {({90}^0-70}^0)=\cos {70}^0$ $(\because \sin ({90}^0-x)=\cos x)$

$\sec {70}^0=\sec {({90}^0-20}^0)=\csc {20}^0$ $(\because \sec ({90}^0-x)=\csc x)$

Therefore, $\sec {70}^0\sin {20}^0-\cos {70}^0\csc {20}^0$ can be evaluated as shown below:

$\sec {70}^0\sin {20}^0-\cos {70}^0\csc {20}^0=\cos {70}^0\csc {20}^0-\cos {70}^0\csc {20}^0=0$

Hence, $\sec {70}^0\sin {20}^0-\cos {70}^0\csc {20}^0=0$.