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Prove that: $(\sin A + \sec A)^2 + (\cos A + \text{cosec } A)^2 = (1 + \sec A \text{ cosec } A)^2$
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Sol. LHS $= (\sin A + \frac{1}{\cos A})^2 + (\cos A + \frac{1}{\sin A})^2$ (1/2 Mark)
$= \sin^2A + \frac{1}{\cos^2 A} + \frac{2 \sin A}{\cos A} + \cos^2A + \frac{1}{\sin^2A} + \frac{2 \cos A}{\sin A}$ (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 Mark)
$= 1 + \frac{\sin^2 A + \cos^2 A}{\cos^2A \sin^2 A} + 2 \frac{\sin^2 A + \cos^2 A}{\cos A \sin A}$ (1/2 Mark)
$= 1 + \sec^2 A \text{ cosec}^2A + 2 \sec A \text{ cosec } A$
$= (1 + \sec A \text{ cosec } A)^2 = \text{RHS}$
$= \sin^2A + \frac{1}{\cos^2 A} + \frac{2 \sin A}{\cos A} + \cos^2A + \frac{1}{\sin^2A} + \frac{2 \cos A}{\sin A}$ (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 Mark)
$= 1 + \frac{\sin^2 A + \cos^2 A}{\cos^2A \sin^2 A} + 2 \frac{\sin^2 A + \cos^2 A}{\cos A \sin A}$ (1/2 Mark)
$= 1 + \sec^2 A \text{ cosec}^2A + 2 \sec A \text{ cosec } A$
$= (1 + \sec A \text{ cosec } A)^2 = \text{RHS}$