214
Prove that :
n$(\sin A + \sec A)^2 + (\cos A + \cosec A)^2 = (1 + \sec A \cosec A)^2$
n$(\sin A + \sec A)^2 + (\cos A + \cosec A)^2 = (1 + \sec A \cosec A)^2$
Show SolutionHide Solution↓
LHS $= (\sin A + \frac{1}{\cos A})^2 + (\cos A + \frac{1}{\sin A})^2$ (1/2 Mark)
$= \sin^2 A + \frac{1}{\cos^2 A} + \frac{2 \sin A}{\cos A} + \cos^2 A + \frac{1}{\sin^2 A} + \frac{2 \cos A}{\sin A}$ (1 Mark)
$= 1 + (\frac{1}{\cos^2 A} + \frac{1}{\sin^2 A}) + 2 (\frac{\sin A}{\cos A} + \frac{\cos A}{\sin A})$
$= 1 + \frac{\sin^2 A + \cos^2 A}{\cos^2 A \sin^2 A} + 2 \frac{\sin^2 A + \cos^2 A}{\cos A \sin A}$ (1 Mark)
$= 1 + \frac{1}{\cos^2 A \sin^2 A} + \frac{2}{\cos A \sin A}$
$= 1 + \sec^2 A \cosec^2 A + 2 \sec A \cosec A$
$= (1 + \sec A \cosec A)^2 = RHS$
$= \sin^2 A + \frac{1}{\cos^2 A} + \frac{2 \sin A}{\cos A} + \cos^2 A + \frac{1}{\sin^2 A} + \frac{2 \cos A}{\sin A}$ (1 Mark)
$= 1 + (\frac{1}{\cos^2 A} + \frac{1}{\sin^2 A}) + 2 (\frac{\sin A}{\cos A} + \frac{\cos A}{\sin A})$
$= 1 + \frac{\sin^2 A + \cos^2 A}{\cos^2 A \sin^2 A} + 2 \frac{\sin^2 A + \cos^2 A}{\cos A \sin A}$ (1 Mark)
$= 1 + \frac{1}{\cos^2 A \sin^2 A} + \frac{2}{\cos A \sin A}$
$= 1 + \sec^2 A \cosec^2 A + 2 \sec A \cosec A$
$= (1 + \sec A \cosec A)^2 = RHS$