Question

Show that ${(\sin A+co\sec A)}^2+{(\cos A+\sec A)}^2=7+{\tan }^{2}A+{\cot }^{2}A$

Answer 1

$LHS=(\sin A+\text{cosec}A{)}^2+(\cos A+\text{sec}A{)}^2$

$={\sin }^{2}A+{\text{cosec}}^{2}A+2\sin A\text{cosec}A+{\cos }^{2}A+{\text{sec}}^{2}A+2\cos A\text{sec}A$

$=({\sin }^{2}A+{\cos }^{2}A)+(1+{\cot }^{2}A)+(1+{\tan }^{2}A)+2+2$

$=7+{\cot }^{2}A+{\tan }^{2}A=RHS$

Hence Proved