LHS $= (\sin A + \sec A)^2 + (\cos A + \csc A)^2$
$$= \sin^2 A + 2\sin A\sec A + \sec^2 A + \cos^2 A + 2\cos A\csc A + \csc^2 A$$
$$= (\sin^2 A + \cos^2 A) + \sec^2 A + \csc^2 A + 2\cdot\frac{\sin A}{\cos A} + 2\cdot\frac{\cos A}{\sin A}$$
$$= 1 + \sec^2 A + \csc^2 A + 2\left(\frac{\sin^2 A + \cos^2 A}{\sin A\cos A}\right)$$
$$= 1 + \sec^2 A + \csc^2 A + \frac{2}{\sin A\cos A}$$
RHS $= (1 + \sec A\csc A)^2 = 1 + 2\sec A\csc A + \sec^2 A\csc^2 A$
$$= 1 + \frac{2}{\sin A\cos A} + \frac{1}{\sin^2 A\cos^2 A}$$
Now, $\sec^2 A + \csc^2 A = \dfrac{1}{\cos^2 A} + \dfrac{1}{\sin^2 A} = \dfrac{\sin^2 A + \cos^2 A}{\sin^2 A\cos^2 A} = \dfrac{1}{\sin^2 A\cos^2 A} = \sec^2 A\csc^2 A$
Therefore, LHS $= 1 + \dfrac{2}{\sin A\cos A} + \sec^2 A\csc^2 A =$ RHS $\quad\blacksquare$
Source: Chapter 8, Section 8.4 Trigonometric Identities
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