Question

Prove that $\frac{\sin {34}^o}{\sin {56}^o}+\frac{\cos {34}^o}{\cos {56}^o}=\sec {34}^o.cosec{34}^o$

Answer 1

$\frac{\sin {34}^o}{\sin {56}^o}+\frac{\cos {34}^o}{\cos {56}^o}=\sec {34}^o\csc {34}^o$

LHS $\frac{\sin {34}^o}{\sin {56}^o}+\frac{\cos {34}^o}{\cos {56}^o}$

$=\frac{\sin {34}^o}{\sin (90-{34}^o)}+\frac{\cos {34}^o}{\cos (90-{34}^o)}$

$=\frac{\sin {34}^o}{\cos {34}^o}+\frac{\cos {34}^o}{\sin {34}^o}\left[\begin{array}{c}\sin (90-\theta )=\cos \theta \\ \cos (90-\theta )=\sin \theta \end{array}\right]$

$=\frac{{\sin }^2{34}^o+{\cos }^2{34}^o}{\cos {34}^o\sin {34}^o}$

$=\frac{1}{\cos {34}^o\sin {34}^o}({\sin }^2\theta +{\cos }^2\theta =1)$

$=\sec {34}^o\csc {34}^o$

=R.H.S