Question

Prove the following statements.
$(\tan \alpha +cosec\beta {)}^2-(\cot \beta -\sec \alpha {)}^2=2\tan \alpha \cot \beta (cosec\alpha +\sec \beta )$.

Answer 1

L.H.S$={(\tan \alpha +\csc \beta )}^2-{(\cot \beta -\sec \alpha )}^2$

$={\tan }^2\alpha +{\csc }^2\beta +2\tan \alpha \csc \beta -{\cot }^2\beta -{\sec }^2\alpha +2\cot \beta \sec \alpha$

$=({\tan }^2\alpha -{\sec }^2\alpha )+({\csc }^2\beta -{\cot }^2\beta )+2\tan \alpha \csc \beta +2\cot \beta \sec \alpha$

$=-1+1+2\tan \alpha \csc \beta +2\cot \beta \sec \alpha$

$=2\tan \alpha \cot \beta (\frac{1}{\sin \beta }\frac{\sin \beta }{\cos \beta }+\frac{1}{\cos \alpha }\frac{\cos \alpha }{\sin \alpha })$

$=2\tan \alpha \cot \beta (\sec \beta +\csc \alpha )$

$=$R.H.S

Hence proved.